1984 AHSME Problems/Problem 9

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Problem

The number of digits in $4^{16}5^{25}$ (when written in the usual base $10$ form) is

$\mathrm{(A) \ }31 \qquad \mathrm{(B) \ }30 \qquad \mathrm{(C) \ } 29 \qquad \mathrm{(D) \ }28 \qquad \mathrm{(E) \ } 27$

Solution

We can rewrite this as $2^{32}5^{25}$. We can also combine some of the factors to introduce factors of $10$, whose digit count is simple to evaluate because it simply adds $0$s. Thus, we have $2^{32}5^{25}=2^72^{25}5^{25}=2^710^{25}$. We can see that this final number is $2^7$ with $25$ $0$s annexed onto it. $2^7=128$, which has $3$ digits, so the entire number has $25+3=28$ digits, $\boxed{\text{D}}$.

See Also

1984 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 8
Followed by
Problem 10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
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