Question

Let K be the largest number with exactly 3 factors that divide 25! How many factors does (k – 1) have?
16
12
9
14

Answer 1

If and only if the total no. of factors of an integer is an odd number, then the integer will be a perfect square.

Here the integer k has exactly 3 factors, so we can say that k is a perfect square, and those three factors will be,

1,  √k,  k

Here, √k should be a prime number.

In 25!, numbers 11 and 22 are multiplied, so there will be 11² = 121 divides 25!.

Only 13 is multiplied in 25! but not 26 and other multiplies of 13. So 13² = 169 does not divide 25!.

Thus we can say that k = 121.

⇒   k - 1 = 120

120 = 2³ × 3 × 5

So the total no. of factors of 120 = (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16.

Hence 16 is the answer.