2026 AMC 8 Problem 24

Question

The notation

n!

(read “

n

factorial”) is defined as the product of the first

n

positive integers. (For example,

3! = 1 \cdot 2 \cdot 3 = 6.

) Define the superfactorial of a positive integer, denoted by

n^!,

to be the product of the factorials of the first

n

integers. (For example,

3^! = 1! \cdot 2! \cdot 3! = 12.

) How many factors of

7

appear in the prime factorization of

51^!,

the superfactorial of

51?

147

150

156

168

171

Count how many times multiples of

7

appear in the factorials

Use

\sum_{n=1}^{51}(\lfloor \frac{n}{7}\rfloor+\lfloor \frac{n}{49}\rfloor)

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For each

n!

, the number of factors of

7

is

\lfloor \frac{n}{7}\rfloor+\lfloor \frac{n}{49}\rfloor

. Therefore the exponent of

7

in

51^{!}

is

\sum_{n=1}^{51}\lfloor \frac{n}{7}\rfloor+\sum_{n=1}^{51}\lfloor \frac{n}{49}\rfloor

. The first sum is

7(1+2+3+4+5+6)+3\cdot7

=168

, and the second sum is

3

. The total is

171

.

Problem 24 in other years