Vincen wrote:What is the remainder when you divide \(2^{200}\) by 7?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

[spoiler]OA=D[/spoiler]

Source: GMAT Club Tests

Note the power cycles of 2.

"¢ 2^1 = 2;
"¢ 2^2 = 4;
"¢ 2^3 = 8;
"¢ 2^4 = 16;

"¢ 2^5 = 32;
"¢ 2^6 = 64;
"¢ 2^7 = 128;
"¢ 2^8 = 256;

We see that units digit has a cyclicity of 4: 2, 4, 8, and 1.

Let's find out the cyclicity of reminders when 2^n is divided by 7.

"¢ 2^1/7 remainder = 2;
"¢ 2^2/7 remainder = 4;
"¢ 2^3/7 remainder = 1;

"¢ 2^4/7 remainder = 2;
"¢ 2^5/7 remainder = 4;
"¢ 2^6/7 remainder = 1;

We see that the cyclicity of reminders when 2^n is divided by 7 is 3.

Note that closest no. to 200 is 198, which is a multiple of 3; we can write 2^200 as 2^198 * 2^2.

Thus, the remainder when you divide \(2^{200}\) by 7 = Remainder of 2^198 * Remainder of 2^2 = Remainder of 2^(3*66) * Remainder of 2^2
= 1 * 4 = 4

The correct answer: D

Hope this helps!

-Jay
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