What is the remainder, after division by 100, of \(7^{10}\)?
A. 1
B. 7
C. 43
D. 49
E. 70
The OA is D
Source: GMAT Prep
What is the remainder, after division by 100, of \(7^{10}\)?
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- Jay@ManhattanReview
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The remainder, after division by 100, of \(7^{10}\) would be the last two digits. For example, 243 divided by 100 would leave a remainder 43.swerve wrote:What is the remainder, after division by 100, of \(7^{10}\)?
A. 1
B. 7
C. 43
D. 49
E. 70
The OA is D
Source: GMAT Prep
Let's observe the units digits of 7^(positive integer)
"¢ 7^1 = 7; units digit = 7;
"¢ 7^2 = 49; units digit = 9;
"¢ 7^3 = 243; units digit = 3;
"¢ 7^4 = _ _ _1; units digit = 1;
"¢ 7^5 = _ _ _7; units digit = 7;
"¢ 7^6 = _ _ _9; units digit = 9;
"¢ 7^7 = _ _ _3; units digit = 3;
"¢ 7^8 = _ _ _1; units digit = 1;
We see that the units digits repeat after every four cycles.
Thus, 7^10 = 7^(2*4 + 2)
The units digit of 7^(2*4 + 2) would be the units digit of 7^2, which is 9. The only option eligible is D.
The correct answer: D
Hope this helps!
-Jay
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- Scott@TargetTestPrep
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7^10 = (7^2)^5 = 49^5 = 49^2 x 49^2 x 49 = 2401 x 2401 x 49. Notice that 2401/100 = 24 R 1, so (2401 x 2401 x 49)/100 has the same remainder when (1 x 1 x 49)/100. However, since 1 x 1 x 49 = 49, so the remainder must be 49.swerve wrote:What is the remainder, after division by 100, of \(7^{10}\)?
A. 1
B. 7
C. 43
D. 49
E. 70
The OA is D
Source: GMAT Prep
Alternate Solution:
Let's find the units digit of 7^10. We look at the pattern of the units digits of powers of 7. When writing out the pattern, notice that we are concerned ONLY with the units digit of 7 raised to each power.
7^1 = 7
7^2 = 9
7^3 = 3
7^4 = 1
7^5 = 7
The pattern of the units digit of powers of 7 repeats every 4 exponents. The pattern is 7-9-3-1. In this pattern, all positive exponents that are multiples of 4 will produce 1 as its units digit. Thus:
7^8 has a units digit of 1, 7^9 has a units digit of 7 and so 7^10 has a units digit of 9. The only answer choice with a units digit of 9 is D.
Answer: D
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