A positive integer x has 60 divisors and 7x has 80 divisors. What is the greatest integer y such that 7y7y divides n? Choose one:  A) 0  B) 1  C) 2  D) 3  E) 4
Total number of factors of x = 60 = (p+1)*(q+1)*(r+1)... = 2^2 * 3 * 5   Now note that 7x has only one 7 more than x. The number of all other prime factors stays the same.   Total number of factors of 7x = 80 = (p+2)*(q+1)*(r+1)... = 2^4 * 5   Here, the 3 of previous expression has disappeared so it must have converted to 4. Does it make sense? Let's see:   Total number of factors of x = 60 = 2^2 * 3 * 5 = (3+1)*(2+1)*(4+1)   Total number of factors of 7x = 80 = 2^2 * 4 * 5 = (3+1)*(3+1)*(4+1)   The powers of other factors stay the same. Only the power of 7 increases by 1. So initially, in x, the power of 7 must have been 2. Hence the maximum value of y must be 2.   Answer (2)
<result>C</result>
A positive integer x has 60 divisors and 7x has 80 divisors. What is the greatest integer y such that 7y7y divides n?