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# The number 25^64*64^25 is the square of a positive integer N. In decim

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Math Expert
Joined: 02 Sep 2009
Posts: 54371
The number 25^64*64^25 is the square of a positive integer N. In decim  [#permalink]

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19 Mar 2019, 00:27
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Question Stats:

42% (02:50) correct 58% (02:06) wrong based on 12 sessions

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The number $$25^{64}*64^{25}$$ is the square of a positive integer N. In decimal representation, the sum of the digits of N is

(A) 7
(B) 14
(C) 21
(D) 28
(E) 35

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The number 25^64*64^25 is the square of a positive integer N. In decim  [#permalink]

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Updated on: 20 Mar 2019, 01:08
1

Solution

Given:
• The number $$25^{64} ∗ 64^{25}$$ is the square of a positive integer N.

To find:
• The sum of the digits of N, in decimal representation.

Approach and Working:
$$N^2 = 25^{64} ∗ 64^{25} = (5^2)^64 * (2^6)^{25} = 5^{128} * 2^{150}$$
• Therefore, $$N = 5^{64} * 2^{75} = 5^{64} * 2^{64} * 2^{11} = 2048 * 10^{64}$$
• So, the sum of the digits of N = 2 + 0 + 4 + 8 = 14

Hence, the correct answer is option B.

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Originally posted by EgmatQuantExpert on 19 Mar 2019, 01:07.
Last edited by EgmatQuantExpert on 20 Mar 2019, 01:08, edited 1 time in total.
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Re: The number 25^64*64^25 is the square of a positive integer N. In decim  [#permalink]

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19 Mar 2019, 05:00
Bunuel wrote:
The number $$25^{64}*64^{25}$$ is the square of a positive integer N. In decimal representation, the sum of the digits of N is

(A) 7
(B) 14
(C) 21
(D) 28
(E) 35

25^64 * 64^ 25
we can write
5^128 * 2^150
squaring
5^64 * 2^75
10^64 * 2^ 11
2^11- 2048 ;sum of digits = 14
IMO B
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Re: The number 25^64*64^25 is the square of a positive integer N. In decim   [#permalink] 19 Mar 2019, 05:00
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# The number 25^64*64^25 is the square of a positive integer N. In decim

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