2002 AMC 10B Problem 14

Question

The number

25^{64} \cdot 64^{25}

is the square of a positive integer

N.

In decimal representation, what is the sum of the digits of

N?

7

14

21

28

35

Take the square root by writing everything as powers of

2

and

5

N = 5^{64}\cdot 2^{75} = 10^{64}\cdot 2^{11}

; only the digits of

2^{11}

matter

Since

25 = 5^2

and

64 = 2^6,

we have

25^{64}\cdot 64^{25} = 5^{128}\cdot 2^{150},

so

N = \sqrt{5^{128}\cdot 2^{150}} = 5^{64}\cdot 2^{75}.

Writing

2^{75} = 2^{64}\cdot 2^{11},

we get

N = (5\cdot 2)^{64}\cdot 2^{11} = 10^{64}\cdot 2048.

So

N

is

2048

followed by

64

zeros, and its digit sum is

2 + 0 + 4 + 8 = 14.

Thus, the correct answer is B.

Problem 14 in other years