2019 AMC 10A Problem 2

Question

What is the hundreds digit of

(20!-15!)?

0

1

2

4

5

Look only at divisibility by

1000

Both factorials contain at least three factors of

5

and three factors of

2

Both

20!

and

15!

contain at least three factors of

5

and three factors of

2

, so both are divisible by

2^3\cdot5^3=1000.

Their difference is therefore also divisible by

1000.

Being a multiple of

1000

makes the last three digits

0,

which shows that the hundreds digit is also

0.

Thus, A is the correct answer.

Problem 2 in other years