If K is a positive integer, what is the remainder when

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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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

To determine the units digit of 13^(4K+2) + 8, we need to determine the units digit of this sum.

If we let K = 1, then we have 13^6, whose units digit is equivalent to the units digit of 3^6. The units-digit pattern of powers of 3 is 3-9-7-1. That is, the units digit is 1 when 3 is raised to a power that is a multiple of 4. Therefore, 3^4 has a units digit of 1, 3^5 has a units digit of 3, and 3^6 has a units digit of 9.

Since 9 + 8 = 17, the remainder when 17/10 is 7.

Answer: A