Category 3 Number Theory Meet #1, October, 2000

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5 Category 3 Meet #1, October, For how many positive integral values of n will 168 n be a whole number? 2. What is the greatest integer that will always divide the product of four consecutive integers? 3. Find the sum of all the positive integers less than 1000 that are both perfect squares and perfect cubes

6 Solutions to Category 3 Meet #1, October, The positive integral values of n that will make 168 n a whole number are the factors of 168. (By positive integral values we mean positive integers or whole numbers.) We can simply list all the factors of 168 in pairs: 1 168, 2 84, 3 56, 4 42, 6 28, 7 24, 8 21, and Those are the 16 values of n that will make 168 a whole number. Alternatively, there is a trick for figuring out the number of factors of a number without actually listing them. We express 168 in prime factors with exponents as follows: 168 = We then raise each exponent by one and multiply them like this: ( 3 + 1) ( 1 + 1) ( 1 + 1) = = 16. n 2. The product of four consecutive integers is guaranteed to contain a multiple of 3, and two multiples of 2, one of which is also a multiple of 4. Being careful not to count the same factor of 2 as both a multiple of 2 and a multiple of 4, we can be certain of only three factors of 2 along with the one factor of 3. From these factors we obtain the product = 24, so 24 will always divide the product of four consecutive integers. Some people will find this answer simply by multiplying the first four counting numbers: = Numbers that are both perfect squares and perfect cubes are sixth powers, because a ( a ) ( a ) = = Since the cube root of 1000 is 10, we need only values under 10 for a 2. Now if a 2 is under 10, then a must be 3 or less. The three sixth powers under 1000 are: =, = =, and 3 6 = 729. Their sum is

7 Category 3 Meet #1, October, What is the greatest possible difference of two primes if both of those primes are between 50 and 100? 2. Use the following clues to find the value of n: n is a whole number n is divisible by 24 n is greater than 300 n is not divisible by 17 n is less than 400 n is divisible by 21 the sum of the digits of n is Find the sum of all the different prime factors of 364,

8 Solutions to Category 3 Meet #1, October, To get the greatest possible difference, we will want to find the prime closest to 100 and subtract the prime closest to = 9 11, so it s not prime, and 51 = 3 17, so it s not prime either. Both 97 and 53 are prime, though, so our greatest difference is 97 53, which is That n is divisible by 24 and 21 are the most 3 helpful clues to start with. Since 24 = 2 3 and 21 = 3 7, we know that n is divisible by 2 3, 3, and 7, which are relatively prime. This means that n is a multiple of = 168. The only multiple of 168 that is between 300 and 400 is = 336, so that must be it. Checking the other clues, we know that 336 cannot be divisible by 17, since we created and 17 is a it by multiplying ( ) different prime. The final clue says that the sum of the digits is 12 and this is true of our answer , 000 = and we know that = 2 5. Now we need to look for other prime factors in 364: = = 2 91 = So the different prime factors of 364,000 are 2, 5, 7, and 13. The sum of these numbers is 27.

9 Category 3 Meet #1, October, If the five-digit number 837A5 is known to be divisible by 15, what is the sum of all the possible values of the digit A? 2. Twin primes are two primes that have a positive difference of 2, such as 11 and 13. What is the sum of the least twin primes greater than 50? 3. Find the value of n, if n is a natural number, n is less than 100, n is the product of two primes, the sum of the digits of n is 10, and the positive difference between the two prime factors of n is a multiple of

10 Solutions to Category 3 Meet #1, October, or To be sure a number is divisible by 15, it must be divisible by both 3 and 5. The five-digit number 837A5 is definitely divisible by 5, since its units digit is a 5. If it is to be divisible by 3, the sum of its digits must be a multiple of 3. The known digits have a sum of = 23. This means that A can have the values 1, 4, or 7. The sum of these possible values is = There are no twin primes in the fifties or the sixties. Thus the least twin primes greater than 50 are 71 and 73. Their sum is 144. Editor note: The original meet had 144 as the answer, but 59 and 61 are also twin primes, so the correct answer should be 120. Thanks to Zhiping You for pointing this out. 3. There are too many possible products of two primes to start with that clue. The natural numbers less than 100 with a sum of digits equal to 10 are 19, 28, 37, 46, 55, 64, 73, 82, and 91. Of these, four of these are the product of two primes: 46 = 2 23, 55 = 5 11, 82 = 2 41, and 91 = Since the difference between the prime factors of 82 is 41 2 = 39 and 39 is a multiple of 13, n must be 82.

11 Category 3 Meet #1, October = 2 a 3 b 5 c 7 d 11 e Find the value of a + b + c + d + e. 2. Find the sum of all possible values of the digit N such that the 5-digit number 318N4 is divisible by Melanie s locker number is the product of the least pair of consecutive primes that have a difference of 6. What is Melanie s locker number?

12 Solutions to Category 3 Meet #1, October = , so a + b + c + d + e = =7. 2. A number that is divisible by 12 must be divisible by both 3 and 4. For the 5-digit number 318N4 to be divisble by 3, the sum of its digits must be a multiple of = 16, so N would have to be 2 or 5 or 8 to make a multiple of 3. This gives us 31824, 31854, and to consider. To determine which among these is divisible by 4, we need only check the last two digits of each number, since any multiple of 100 is divisble by and 84 are multiples of 4, but 54 is not. Thus only and are divisible by 12 and the sum of all possible values of N is = The least pair of consecutive primes with a difference of 6 is the pair 23 and 29. Their product is 667, so Melanie s locker number must be 667

13 Category 3 Meet #1, October How many pairs of primes have a sum of 24? 2. If the five-digit number 5N82N is divisible by 18, what is the value of N? 3. How many of the positive factors of 660 are odd?

14 Solutions to Category 3 Meet #1, October There are three (3) pairs of primes that have a sum of 24. They are: , , and For the five-digit number 5N82N to be divisible by 18, it must pass the divisibility tests for both 2 and 9. Since there is an N in the units place (ones place), we know that N has to be even. To be divisible by 9, the sum of the digits must be a multiple of 9. Currently the sum of the digits is 5 + N N = N. For N = 0, we get = 15, which is not a multiple of 9. For N = 2, we get = 19. For N = 4, we get = 23. For N = 6, we get = 27, which is a multiple of 9. This tells us that the five-digit number is divisible by 18. ( = 3157) Note that this is the only single-digit value of N that will make N equal a multiple of 9. Thus, N = The factor pairs of 660 are listed below. The eight (8) odd factors are in bold

15 Category 3 Meet #1, October Two prime numbers, each less than 50, have a difference of 35. What is the sum of these two primes? 2. The letters A and B each represent one digit in the five-digit number AB9BA. If AB9BA is divisible by 2, 3, 4, 6, 8, and 9, what is the value of 2A + B? 3. How many whole numbers between 80 and 100 are the product of two distinct primes? In other words, how many whole numbers between 80 and 100 have exactly four factors?

16 Solutions to Category 3 Meet #1, October After 2, all primes are odd. Any two odd primes will have an even difference. If the difference of two primes is odd, then 2 must be one of the primes. Since our difference is 35, the other prime must be = 37. Our two primes are 2 and 37 and their sum is = The sum A + B B + A must be a multiple of 9, since the five-digit number AB9BA is divisible by 9. This means that A + B = 9. Also, A must be even, since AB9BA is a divisible by 2. The possibilities for the last two digits BA are 18, 36, 54, 72, and 90. Only 36 and 72 are divisible by 4, so the 5-digit number is either or Since 936 is divisible by 8 and 972 is not, the number must be So A = 6 and B = 3, and the value of 2A + B is = The prime factorizations of all the numbers between 80 and 100 are listed below. The 8 that are products of two primes are in bold. 81 = = = = = = = = = = = = = = = = = = =

17 Category 3 Meet #1, October What is the least three-digit number that is divisible by exactly three different prime numbers? 2. The five-digit number 35N2N is divisible by 2, 3, 4, 6, 8, 9, and 12, among other numbers. What is the value of the digit N? 3. How much greater is the product of the first four primes than the product of the first three composites?

18 Solutions to Category 3 Meet #1, October 2006 Average score: 1.5 answers correct Some Incorrect Seen , 110, 120, Some people might be fooled by = 105, but = 102 is the least three-digit number that is divisible by exactly three primes. 2. N clearly must be an even digit, so 0, 2, 4, 6, and 8 are possible. For 35N2N to be divisible by 3, the sum of the digits must be a multiple of three. So far we have N N = N. Let s use the possible values of N to evaluate this expression. If N = 0, then N = = 10, which is not divisible by 3. If N = 2, then N = = 14, which is not divisible by 3. If N = 4, then N = = 18, which is divisible by 3. Neither N = 6 nor N = 8 gives a multiple of 3, so 4 is our only candidate. Let s make sure that the number is divisible by all the other numbers listed. It is divisible by 4 since the last two digits 24 are divisible by 4. It is divisible by 6 since it passes the test for 2 and 3. It is divisible by 8 since the last three digits 424 are divisible by 8. It is divisible by 9 since the sum of the digits (18) is divisible by 9. Finally, it is divisible by 12 since it passes the test for 3 and 4. The digit N is indeed The product of the first four primes is = 210. The product of the first three composites is = 192. The difference is = , 6