## Thursday, 2 July 2015

1. If x and y are positive integers such that x is a factor of 10 and y is a factor of 12, all of the following  could be the value of xy EXCEPT
(A) 1  (B) 4  (C) 15  (D) 36  (E) 40
Solution: (D)
If x is a factor of 10, x could equal 1,2,5 and 10. I f y is a factor of 12, y could equal 1,2,3,4,6 and 12.  Choice (A) is the result of multiplying 1 and 1. Choice (B) is the result of multiplying 1 and 4 or 2 and  2.Choice (C) is the result of multiplying 5 and 3 .Choice (E) is the result of multiplying 10 and 4. The  answer is choice (D).

2. How many distinct prime factors does 14! Have?
(A) 2  (B) 5  (C) 6  (D) 7  (E) 9
Solution: (C)

14! equals 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7x 6x 5x 4 x 3 x 2x 1. Its distinct prime factors are the prime  numbers from 1 to 14, which are 2,3,5,7,11 and 13. The answer is choice (C).

3. If the positive integers a and b are distinct factors of 9, then the smallest possible value for ab is
(A) 1  (B) 3  (C) 9  (D) 81  (E) 162
Solution (B)
The distinct factors of 9 are 1, 3 and 9. The smallest product of any two of these is 1 x 3 = 3. The answer is choice (B).

4. If integer x has at least three distinct prime factors, then which of the following could be an  integer?
(A) 3/x  (B) 15/x  (C) 24/x  (D) 35/x  (E) 42/x
Solution: (E)
For a number divided by x to be an integer, that number must share the factors of x. Choice (A) has only  one prime factor, 3.Choice (B) has only two prime factors, 3 and 5. Choice (C) has only two distinct prime  factors, 2 and 3. Choice (D) has only two prime factors, 5 and 7. Choice ( E) has three prime factors, 2,3
and 7. The answer is choice (E).

5. The product of distinct integers a,b and c is 1,001. IF a,b and c are each greater then 1, what is the smallest value any one of the integers can have?
(A) 3  (B) 7  (C) 11  (D) 13  (E) 77
Solution: (B)
The prime factorization of 1,001 is 7 x 11 x 13 , so each integer a,b and c must equal one of these values. The smallest value any of the integers can have, then is 7. The answer is choice (B).