1977 AMC 12 Problem 25

Attempt Problem 25 of the 1977 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1977 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

25.

Determine the largest positive integer

n

such that

1005!

is divisible by

10^n.

102

112

249

502

none of these

Count the factors of

5

1005!

Include contributions from multiples of

25,

125,

and

625

There are more factors of

2

than of

5,

so the exponent of

10

\begin{aligned} n={}&\left\lfloor\frac{1005}{5}\right\rfloor+ \left\lfloor\frac{1005}{25}\right\rfloor\\ &+\left\lfloor\frac{1005}{125}\right\rfloor+ \left\lfloor\frac{1005}{625}\right\rfloor\\ ={}&201+40+8+1=250. \end{aligned}

Since

250

is not listed, the correct choice is “none of these.”

Therefore, the correct answer is E.

Problem 25 in other years