Math 41 Practice for Test 4

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1 Math 41 Practice for Test 4 1. In a survey, 5 out of 0 fourth graders said they liked their music class. The rest said they did not. a) Among those surveyed, what is the ratio of students who like their music class to those who do not? b) Among those surveyed there are times as many who like their music class as those who do not. c) Among those surveyed there are times as many who do not like their music class as those who do. d) What fraction of the students interviewed like their music class? e) What percent of the students interviewed don t like their music class?. The ages of a mother and a daughter are 50 years and 0 years. a) Make an additive comparison of their ages b) Make a multiplicative comparison of their ages.. A mathematician tells you that the ratio of museums to parks in a town is 0. What would you conclude? 4. Perry is offered two jobs. Job #1 pays $500 for a 40-hour week while Job # pays $450 for a 6- hour week. Which job has a better pay rate? Explain how you decide. 5. Meg paid $1 for ¾ pound of crab. What is the price per pound? a) Show how to answer this question using a part-whole diagram. b) Show how to answer this question using a proportion. 6. Two college students, Abe and George, were hired to paint the floor in the gym. Abe, who worked more slowly, painted only /5 as much of the floor as George did. a) Mark the drawing of the floor to show the amounts of floor painted by each of the students. b) Abe s part is how many times as much as George s part? c) What part of the floor did Abe paint? d) What is the ratio of the part painted by George to the part painted by Abe? e) If they are paid $500 to paint the floor, what would be a fair split of the $500? 7. At a picnic, one table seating 10 people has platters of food. A second table seating 14 people has 4 platters of food. Which table is allotting more food per person? Explain your answer using fractions rather than decimals. 8. Use a rectangle to represent a submarine sandwich. Divide the sandwich into two sections A and B so that the ratio of A to B is :4. a) What fraction of the whole sandwich is section A? b) Section B is how many times as long as section A? c) Section A is how many times as long as section B? 1

2 9. One day, Kai and Mai are discussing their ages. They were born on the same day but not the same year. Kai says, Today I m 5 years older than you are. Mai says, Yes, and 4 years ago you were 50% older than I was then. What will be the ratio of Mai s age to Kai s age, 6 years from today? Show your work. 10. Three cities, Avalon, Bayside and Clarkville, agree to share the cost of a park that will be bordered by the three cities. Because their populations are different, they agree that Bayside will pay 1¼ as much as Avalon and Clarkville will pay half as much as Bayside pays. Solve this problem without using algebra. Instead work with diagrams and/or quantitative reasoning. a) What fractional part of the whole cost will each city pay? b) What is the ratio of Avalon s share to Clarkville s share? 11. Re-write each of the given forms in the specified form. a) Given: Quantity P is ½ times as large as Quantity R. Re-write in the ratio form P:R = b) Given: M:N = 7:1 Re-write as Quantity N is times a s much as Quantity M. c) Given: Quantity N is 60% more than Quantity P Re-write in the ratio form N:P = 1. a) Written as a decimal, 0.65% = ; as a fraction, 0.65% =. b) Written as a percent, 6.5 = ; as a fraction, 6.5 =. c) Written as a decimal, 6.5% = ; as a fraction, 6.5% =. 1. Two brothers, Larry and Moe, were given some candy by their older brother, Curly. Larry and Moe are upset. Larry says, I only got ¾ as much candy as you did, Curly. Moe says, Me too! What part of all the candy did Curly get? 14. In the year 000, Smalltown s budget was $100 million. In 001, the budget figure was 0% higher than the year 000 budget and the 00 budget was 10% higher than the 001 budget. What was the 00 budget amount in millions of dollars? 15. One recipe to serve 8 people calls for cups of rice. Since Al expects only 6 people to come for dinner, he decides to use 1 cup of rice. a) Explain Al s thinking. b) Give another way to decide how much rice is needed for 6 people. c) Which way is better, your way or Al s way? Why? 16. a) Is 7:4 the same as 6:? Why or why not? b) Is 4: the same as 8:6? Why or why not? 17. The Math Department uses tablespoons of coffee with 5 cups of water to make coffee. The Science department uses 5 tablespoons of coffee with 9 cups of water to make coffee. Which coffee is stronger? Explain how you decide. 18. The big dog weighs 5 times as much as the little dog. The little dog weighs /8 as much as the medium size dog. The medium sized dog weighs 10 pounds more than the little dog. How much does the big dog weigh?

3 19. Adam and Matt are using different city road maps. On Adam s map, a line 6 inches long represents a road that is really 9 miles long. On Matt s map, a line 8 inches long represents a distance that is really 1 miles long. If both boys were to measure the distance from City A to City B along a line, which would have the longer line in terms of inches? Explain how you decide. 0. Two workers spent 8 hours making a total of 16 parts. Worker A makes 1 parts in one hour. If the workers work at a steady rate throughout the day, who is more productive - Worker A or Worker B? 1. Solve each proportion. Which, if any, can be solved easily without using the cross multiplication property? 4 n 4 n 5 1 a) = b) = c) = d) n = e) = n 10 9 n 15. Use a table to solve the problem: A candy store sells 4 pounds of chocolate for $5.60. How much would 6 ¾ pounds of chocolate cost?. If the box shown below represents 75% of an amount, show a box that represents 15% of the same amount and one that represents 50% of the same amount. 4. The decrease in the price of notebooks was $1.05 which was a 0% discount. What were the original price and the new price? 5. The population of City A is 15,000 while the population of City B is 0,000. a) The population of City A is what percent of the population of City B? b) The population of City B is what percent of the population of City A? c) The population of City A is what percent smaller than the population of City B? d) The population of City B is what percent larger than the population of City A? 6. The police chief said, Accidents are down in our town, with about 80% as many this year as last year. This year there were 75 accidents. If the police chief was correct, how many accidents were there last year? 7. Show how to estimate each of the following and, if possible, predict (without actually calculating) whether your estimate will be larger or smaller than the actual value: a) 49% of 15 people b) 15% of $9.15 c) 76% of 99 miles d) is 51% of what? e) 1 is what % of 79?

4 8. If the platter of cookies shown below contains /10 of a box of cookies, how many cookies were in the box? 9. a) Write an equation that states that 16 is a factor of the number k. b) Write an equation that states that m is a factor of n. 0. Explain why each of the following is a composite number: 15, 6, Complete the following addition and multiplication tables for even and odd numbers. + even odd x even odd even even odd odd Can you make any definite statements about: a) The sum of any number of even numbers? b) The sum of any number of odd numbers? c) The product of any number of even numbers? d) The product of any number of odd numbers?. If 5 is factor on a number n, give two other factors of n besides 1 and n.. How many factors does each of the following have? 4 6 a) b) i i 5 c) 5711 i i i d) Consider the number m = 8 i1 i 1 7. Which of the following could not be factors of the number m. Explain how you know. 7 a) i 11 8 b) i 1 5. The number 57,79,64,596 has too many digits for most calculators to display. Use divisibility tests to decide which of the numbers,, 4, 5, 6, 8, 9, 10 and 11 divide this number. 6. Create a 6-digit number such that a) and are factors of the number but 9 is not. b) is a factor of the number but 4 is not c) 8 and 9 are factors of the number 7. Given the number 57,4, use divisibility tests to find all possible numbers that, when placed in the, make the number divisible by each of the following numbers: a) b) c) 4 d) 5 e) 6 f) 8 g) 9 h) 10 i) 11 4

5 9. What, if anything, can you say about the oddness or evenness of m if a) 5,06,8 x m is an even number b) 5,06,8 + m is an even number 40. If n = , then is n an even number, or an odd number? Explain your answer. 41. Circle T if the statement is true, F otherwise. a) T F Every whole number is a multiple of itself. b) T F It is possible for an even number to have an odd factor. c) T F Zero is a multiple of every whole number. d) T F 50 is a factor of a) T F There are no values of b and c for which 7 b = 9 c. b) T F There are no values of r and s for which 11 r = 9 s. 4. Is there a whole number M that would make this true? If so, identify the number M, If not, explain why not a) i5 i17 = i17 i M b) i7 i11 i = i7i11 i M 44. If a = i 4 i 11 and b = ii5 6 i 7 find LCM(a,b) and GCF(a,b). Express your answers in factored form. 45. Use factor lists the set intersection method to find LCM(5,56 ) and GCF( 5,56 ). 46. Use prime factorization to find LCM(5,56 ) and GCF( 5,56 ). 47. State a divisibility test for 4 and explain why it works. 48. State a divisibilty test for and explain why it works. ANSWERS 1. a) 5:1 b) 5 c) 1 d) 5 e) a) Mom is 0 years older than her daughter. b) Mom is times as old as her daughter.. If the ratio of museums to parks is 0, there are no museums in town b/c 0 = 0 where n is any nonzero number representing the number of parks. This ratio can be 0 if and only if the numerator is 0. n Amount paid $ 4. Job #1 pays $500 for 40 hours which means $1.50 per hours since = 500 = # hours 40hours Likewise, Job # pays $450 per hour or $1.50 per hour. Pay rates are the same. 5. a) part-whole diagram model Meg pays $1 for ¾ of a pound of lobster. Let the rectangle represent 1 pound of lobster. Subdivide into 4 parts. The first parts (¾ of a pound of lobster) cost $1, so each ¼ pound of lobster costs $4. This means that the full pound costs 4 x $4 or $16. $1 for ¾ pound 5

6 b) proportion model here s one possible set-up Amount of lobster > 4 = 1 price paid $ x 1 and solve to get x = $16 6. a) Since Abe painted only /5 as much floor as George did, Abe painted parts for every 5 parts painted by George. b) c) d) 5 e) $ to Abe, $1.50 to George 5 8 Abe s part George s part 7. Table with platters for 10 people allows people allows = platter per person. Since = people provides just a little more food per person. 8. a) Section A is of the whole sandwich. 7 1 b) Section B is 1 times as long as Section A. c) Section A is as long as Section B. 4 platter per person while table with 4 platters for 14 and 0 =, the table with platters for It s important to note that this problem involves both additive and multiplicative comparisons. Kai s statement the Today I m 5 years older you (Mai) are is an additive comparison. She will always be 5 years older than Mai. Mai s statement that four years ago Kai was 50% older is a multiplicative comparison, saying that at that time Kai was half again as old as Mai or Kai was 1½ times as old as Mai. Since we know the difference in their ages is always 5 years we can determine that 50% of Mai s age four years ago was 5 years which means her age then was 10 years, making Kai 15 years old. Today, the girls are 4 years older making Mai 14 and Kai 19. Six years from now, Mai will be 0 and Kai will be 5 so that the ration of Mai s age to Kai s age then will be 5:0 or 5:4. Ages 4 years ago Mai Kai 5 yrs 50% of Mai s age 4 yrs ago 10. Bayside pays 1¼ times as much as Avalon. Clarkville pays half as much as Bayside. Cutting each of Avalon s and Bayside s payments in two and dividing Clarkville s payment into five provides a compatible way to measure the payments. There are equal parts! Avalon Bayside Clarkville Avalon Bayside Clarkville a) Avalon pays 8 10 of the cost while Bayside pays b) The ratio of Avalon s share to Clarkville s share is 8: a) P:R = ½ : 1 or P:R = 7: b) 1/7 c) N:P = 8:5 6

7 a) 0.065% = ; 0.065% = or b) 6.5 = 65%; 6.5 = c) 6.5% = 0.065; 6.5% = or Larry and Moe each received ¾ as much candy as Curly. This means that for each 4 candies Curly 4 got, Larry and moe each received pieces. It looks like this and Curly gets or of the candy. Curly Larry Moe x x x x x x x x x x Year 000 budget was $100 million. Year 001 budget was 0% nigher than year 000 budget so it was 10% of year 000 budget or $10 million. Year 00 budget was 10% higher than year 001 budget so year 00 was 110% of year 001 budget or $1 million. 15. a) Al though if there were fewer people he needed fewer cups of rice. b) Another way to decide is to use a proportion comparing number of cups of rice to number of x people being served. Use = where x = the number of cups needed for 6 people and 8 6 x = ¼ cups c) Proportional reasoning works better. If 8 peope need cups of rice, 4 people would need half as much so Al s recommendation of 1 cup for 6 people makes no sense. 16. a) Although 7 is greater than 4 and 6 is greater than, the rations 7:4 and 6: are not equal because 6: means for every 6 is one set you have in another set as illustrated below. For each 6 stars, there are circles. You could also show this as stars for every 1 circle. If you had 7 stars, you would need an extra ½ circle, not an extra whole circle. b) 4: is the same as 8:6. You can use a diagram argument similar to the one above. 17. Math Department uses Tbsp coffee with 5 cups of water or /5 Tbsp coffee per 1 cup water. Science Department uses 5 Tbsp coffee with 9 cups water or 5/9 Tbsp coffee per 1 cup water. Both rates are slightly larger than ½ Tbsp coffee per 1 cup water. For easy comparison, you can rewrite both fractions with the common denominator 45. /5 = 7/45 while 5/9 = 5/45. The Math Department makes the stronger coffee. Another way to compar is to note the /5 = 6/10 which is 1/10 bigger than ½ while 5/9 = 10/18 which is 1/18 bigger than ½. Again we see that the Math Department coffee is stronger b/c 1/10 >1/ Big dog weights 5 times as much as little dog. DONE IN CLASS! Little dog weighs /8 as much as medium dog. Medium dog weights 10 pounds more that little dog. Big dog weighs 0 pounds. 7

8 19. Adam s map use 6 inches to represent 9 miles so that each inch on Adam s map covers 1½ miles. On Matt s map, 8 inches represents 1 miles so that Matt s map also uses the scale 1 inch per 1½ miles. If both boys measured the distance from City A to City B along a line, they would both have the same measurement in inches. 0. Worker B is more productive. She makes 14 parts per hour. 1. a) n = 1 5 b) n = 1 c) n = 6 d) n = 70 9 e) n = 5. 4 pounds of candy costs $5.60. Find cost of 6 ¾ pounds using a table. # pounds cost comments 4 $5.60 $.80 ½ the cost of 4 pounds 1 $1.40 ½ the cost of pounds ½ $0.70 ½ the cost of 1 pound ¼ $0.5 ½ the cost of ½ pound 6 ¾ $10.85 Total pounds & cost. Box shown here represents 75% of an amount. Subdivide into three parts, each 5% Use this information to make boxes showing 15% of the same amount >>>>> 50% of the same amount >>>>>>. 4. Original price was $.50; new price is $ a) 75% b) 1 1 % c) 5% d) 1 % 6. Accidents were about 80% as many this year as last year and there were 75 accidents this year. 75 accidents this year = 80% of last year s total : Accidents this year: Accidents last year: 75 4 = so about x last year 8

9 7. a) 49% of 15 people is about ½ of 15 people = 6.5 but since we re talking people and 49% < ½, we ll use 6. Since 1% of 15 people is 1.5 the actual value of 49% of 15 will be 1.5 less than 6.5 so our estimate of 6 will be larger than the calculated value of 49% of 15 b) 15% of $9.15 is about 10% of $9 plus 5% of $9. So that $.90 + $1.95 = $5.85. This estimate would be slightly smaller than the true value b/c we dropped the 15 cents in the estimate. Or you could estimate that $9.15 is a little smaller that $40 so 15% of $9.15 would be a little less than 15% of $40 which is $4 + $ = $6. This estimate will be larger than the actual value. c) 76% of 99 miles will be close to 75% or ¾ of 400 which is 00. Although you wouldn t do this with young children, for our own understanding we can show that the estimate of 75% of 400 is actually less than the actual value of 76% of 99. Here s the argument, using the distributive property. It shows that the estimate is.4 less than the actual value. Notice that we did not need to calculate 76% of 99 to show this conclusion! 75% of 400 = ( 0. 75) ( 400) = ( ) ( 99+ 1) = ( 076. ) ( 99) + ( 076. ) ( 1) ( 001. ) ( 99) ( 001. ) ( 1) = ( 0. 76) ( 99) = ( ) ( ) d) We can estimate the answer to the question is 51% of what? by noting that is 50% of 64. Our estimated answer of 64 will be a little large b/c we re looking for a number for which is a little more than half the number. This means that half the desired number must be less than so that the desired number is times a value less than and thus must be less than 64. e) We can estimate the answer to 1 is what % of 79? by thinking about the question 0 is what % of 80? We know the answer to this simpler question is 5%. It s not difficult to see that 1 is actually 5% of 84 since 1 x 4 = 84. Since we re looking at 1 as a percent of 79, which is 1 1 smaller than 84, we know from our work with the relative size of fractions that > since each fraction involves the same number of parts (numerator = 1) but the sizes of the parts differ 1 1 ( > since the unit fraction with the smaller denominator is the larger unit fraction) NOTE: There was a cookie missing from the platter! The platter contains 1 cookies which amounts to 1 of the box. This means that of the box would be 1 of 1 cookies or 7 cookies. The whole box would contain 10 times this amount or 70 cookies. 9. a) 16 p = k where p is some whole number. b) m p = n where p is some whole number. 0. For some reason, there was no problem 0 on the sheet. To avoid confusion, I ll wait to renumber until next semester is a composite number because it has factors of and 5 as well as 1 and is a composite number b/c in addition to factors of 6 and 1 it s easy to see that is a factor. 98 is a composite number b/c in addition to factors of 98 and 1 it s easy to see that is a factor. 9

10 . Completed tables are shown below: + even odd x even odd even EVEN ODD even EVEN EVEN odd ODD EVEN odd EVEN ODD a) The sum of any number of even numbers must be even b/c the sum of two even numbers is even which means that each time another even number is added to the sum the result will be an even number.. b) The sum of any number of odd numbers may be even or odd depending on the quantity of numbers added. If an EVEN number of odd numbers is added, the sum will be even since the odd numbers can be paired. Each odd number will have a partner and each pair of odd numbers added produces an even result. If an ODD number of odd numbers is added, the sum will be odd since all but one of the odd numbers can be paired. The sum of the paired odd numbers will be even but the adding the final remaining odd number gives an odd result. c) The product of any number of even numbers must be even b/c if even one of the multipliers is even the product will automatically have a factor of which makes the product even. d) The product of any number of odd numbers must be odd b/c the product of the first two odd factors is odd. Likewise, every other factor is odd so the continued multiplation produces an odd results. There must be at least one even factor to make the product even.. If 5 is a factor of a number n, so are 5 and 7 since they are factors of a) 4 has five factors: 0, 1,,, 4. Using the rule in the text, since the highest power of that appears is the 4 th power and is the only prime factor, the total number of factors is = 5 6 b) i i 5 has 4 factors. 6 is a factor of the number, so each of 7 powers 0, 1,,, 4, 5, 6 is also a factor. is a factor of the number, so each of powers 0, 1, is also a factor. 5 1 is a factor of the number, so each of powers 5 0, 5 1 is also a factor. Any of the powers of can be matched withany of the powers of 5 to create x = 6 factors involving only powers of and 5: = 1, = 5, =, = 15, 5 0 = 9, 5 = 5. Now each of these 6 factors can be matched with the 7 powers of to create 6 x 7 = 4 factors. A list is included below so you can visualize c) 5711 i i i has x x x = 16 factors. d) 50 = and has ( + 1)( + 1)(1 + 1)(1 + 1) = (4)()()() = 48 factors. 10

11 5. a) b) 7 i 11 cannot be a factor of m = 8 i1 i 1 7 b/c 11 is not a factor of m 8 i 1 is definitely a factor of m = 8 i1 i b/c i1 = i1 i 1 is a factor 6. Use divisibility tests to investigate which of the numbers,, 4, 5, 6, 8, 9, 10, and 11 are factors of 57,79,64,58. (I made a typo on this problem. Sorry!!!) is NOT a factor b/c the number is not even. Note that if is not a factor, neither is any other even number. This means 4, 6, 8 and 10 are NOT factors of 57,79,64,58. The sum of the digits of this number is 56 which is not divisible by either or 9 so and 9 are also NOT factors of 57,79,64,58. 5 is not a factor b/c the number does not end in a 5 or a 0. Use divisibility tests to investigate which of the numbers,, 4, 5, 6, 8, 9, 10, and 11 are factors of 57,79,64,57. is a factor b/c the number is even is a factor b/c the sum of the digits is 57and is a factor of is a factor b/c 4 is a factor of the number formed by the last two digits, 7. 5 is not a factor b/c the number doen t end in a 5 or in a 0. 6 is a factor b/c both and are factors. 8 is not a factor b/c 8 is not a factor of the number formed by the last three digits, is not a factor b/c the sum of the digits is 57 and 9 is not a factor of is not a factor b/c the number does not end in a Given the number 57,4, use divisibility tests to find all possible numbers that, when placed in the, make the number divisible by each of the following numbers: a) b) c) 4 d) 5 e) 6 f) 8 g) 9 h) 10 i) 11 a) Since this number ends in (which means it must be an even number), any of the numbers 0,1,,,4,5,6,7,8,9 can be placed in the box and the number is still divisible by. b) The sum of the digits here is = 0+. Filling the box with 1, 4 or 7 makes the sum of the digits, respectively, 1, 4 or 7, all of which are divisible by. c) The number formed by the last two digits is. Filling the box with 1,, 5, 7 or 9 makes the last two digits a number divisible by 4 and consequently makes the given number divisible by 4. d) No matter what you put in the box, this number can t be divisible by 5 b/c it doesn t end in 5 or 0. e) The number must be divisble by both and in order to be divisible by 6. All digits work for divisibility by but only 1, 4, and 7 give divisibility by. The three numbers also make the given number divisible by 6. f) The number formed by the last three digits is 4. Filling the box with or 7 makes the last three digits a number divisible by 8 and consequently makes the given number divisible by 8. g) The sum of the digits here is = 0+. Filling the box with 7 makes the sum of the digits 7, all of which are divisible by 9. h) No matter what you put in the box, this number can t be divisible by 5 b/c it doesn t end in a) If 5,06,8 x m is an even number, m may be either even or odd. We know that is a factor of 5.06,8, so is also a factor of any product involving this number. b) If 5,06,8 + m is an even number, m must be even. If m were odd, the sum would also be odd. 40. If n = , n must be an even number. Notice that is an odd number, so that squaring it gives (odd)(odd) = odd. Adding this odd square to the first addedn which is also odd gives (odd) + (odd) = even for the number n. 11

12 41. a) True b) True c) True d) True 4. a) False. Since 7 b = ( ) b = b while 9 c = ( ) c = c, it follows that 7 b = 9 c if b = c or b = c. Some possible values for b and c are b=6, c=9 or b=1, c=18 or b=18, c=7. etc. b) True, since 11 and 9 have no common factors a) There is no number M that will make i5 i17 = i17 i M. If M included factors 5 i the powers 4 of and 5 would match. However, the rightside of the expression includes a factor of 17 while the left side includes only 17 and we have no way to include something in M (which must be a whole number) that will decrease the power of 17 on the right side b) The statement i7 i11 i = i7i11 i M will be true if M = 711 i since i7 i11 i = i7i7i11 i11 i11 i ( ) 5 6 = i7i11 i 7i If a = i 4 i 11 and b = i i i, then LCM(a,b) = 4 6 i i5 i7 i 11 and GCF(a,b) = 45. Use factor lists the set intersection method to find LCM(5,56 ) and GCF( 5,56 ) Set of multiples of 5 = {5, 70, 10, 140, 175, 10, 45, 80, 15, } Set of multiples of 56 = {56, 11, 168, 4, 80, 6, 9, } So LCM(5, 56) = 80 Set of factors of 5 = {1, 5, 7, 5} Set of factors of 56 = {1,, 4, 7, 8, 14, 8, 56} So GCF(5,56) = 7 i. 46. Use prime factorization to find LCM(5,56 ) and GCF( 5,56 ). 5 = 5i 7 and 56 = i 7 so LCM(5,56) = i5i 7 = 80 You need to use the greatest power of all prime factors involved and GCF(5,56) = 7. The only common factor is 7 and it appears once only in each number. 47. To check for divisibilty by 4, consider the number formed by the last two digits of the given number. If this last two digits number is divisible by 4, so is the number itself. This works b/c we can rewrite any number of three or more digits as the sum of the hundreds + the tens + the units. Since 100 is divisible by 4, any multiple of 100 is also divisible by 4. Consequently, to check if the complete number is divisible by 4, we need only check if the number that is formed in the tens and the units place is divisible by 4. Ex: 546 = = 5(100) is divisible by 4 but 46 isn t, so 546 is not divisible by 4 Ex: 978 = = 97(100) is divisible by 4 and so is 8, so 978 is divisible by If the sum of the digits of a number is divisible, the number itself is divisible by three. Our place value number system use powers of 10 and conveniently, each power of 10 is one larger than a multiple of 9 which makes this rule work. Suppose ABCD represents a base ten number. In expanded form, we would write ABCD = ( 1000)A + ( 100)B + ( 10)C + ( 1)D Rewriting each power of 10 as a multiple of 9 plus 1 and using the distributive property, we see that since divides = ( )A + ( )B + ( 9 + 1)C + ( 1)D 9 it also divides 999A, 99B and 99C. If divides = ( 999)A + ( 99)B + ( 9)C + ()A 1 + ()B 1 + ()C 1 + ()D 1 A+B+C+D, then the entire number is divisible by! 1

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