Ratios

shankar.ashwin wrote:If a school has 3003 students- boys and girls inclusive, which of the following cannot represent the ratio of number of boys to girls.

A) 10:11
B) 16:17
C) 18:19
D) 19:20

If, say, the ratio of boys to girls is 10 to 11, that means you can divide up all of the people into groups of 21, where each group contains 10 boys and 11 girls. So the ratio of boys to girls can only be 10 to 11 if the number of boys is a multiple of 10, the number of girls is a multiple of 11, and the number of students is a multiple of 21. Similarly, the ratio can only be 16 to 17 if the total number of students is divisible by 33, can only be 18 to 19 if the total is divisible by 37, and can only be 19 to 20 if the total is divisible by 39.

So this is just a divisibility problem. If we prime factorize 3003 = 3*1001 = 3*7*143 = 3*7*11*13, we can see that the total is divisible by 21 = 3*7, by 33 = 3*11, and by 39 = 3*13, but not by 37 so the answer is C.