BTGmoderatorLU wrote:Source: GMAT Prep
If K is a positive integer, what is the remainder when \(13^{4K+2}+8\) is divided by 10?
A. 7
B. 4
C. 2
D. 1
E. 0
The OA is A
Note that the units digit of the number \(13^{4K+2}+8\) will decide the remainder when \(13^{4K+2}+8\) is divided by 10.
The number \(13^{4K+2}\) can be written as \(13^{4K} \times 13^2\).
The units digit of \(13^{4K}\) and \(13^2\) will be determined by the units digit of \(3^{4K}\) and \(3^2\). The units digit of \(3^{4K}\) has cyclicity of 4, i.e. after every four digits, its cycle is repeated. Let's see below.
"¢ \(3^1\) = 3; Units digit is 3;
"¢ \(3^2\) = 9; Units digit is 9;
"¢ \(3^3\) = 27; Units digit is 7;
"¢ \(3^4\) = 81; Units digit is 1;
Units digits are in order of 3, 9, 7 and 1.
"¢ \(3^5\) = 243; Units digit is 3;
"¢ \(3^6\) = ...9; Units digit is 9;
"¢ \(3^7\) = ...7; Units digit is 7;
"¢ \(3^8\) = ...1; Units digit is 1;
Thus, the units digits of \(13^{4K}\) and \(13^2\) are 1 and 9, respectively. Thus, the units digit of \(13^{4K+2}\) is 1*9 = 9. Thus, the units digit of \(13^{4K+2}+8\) is 9 + 8 = 17. Thus, the remainder when \(13^{4K+2}+8\) is divided by 10 is the remainder when 17 is divided by 10, i.e. 7.
The correct answer:
A
Hope this helps!
-Jay
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