Note the power cycles of 2.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
"¢ 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
_________________
Manhattan Review GRE Prep
Locations: GRE Classes Ho Chi Minh City | GRE Prep Course Kuala Lumpur | GRE Prep Shanghai | SAT Prep Course Hong Kong | and many more...
Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.
