MathRevolution wrote:

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Math Revolution GMAT math practice question]

What is the remainder when \(7^8\) is divided by \(100\)?

A. \(1\)

B. \(2\)

C. \(3\)

D. \(4\)

E. \(5\)

When an integer is divided by 100, the remainder will have the same units digit as the integer.

Thus, to determine which answer choice represents the remainder when \(7^8\) is divided by 100, we need to know the units digit of \(7^8\).

When an integer is raised to consecutive powers, the resulting units digits repeat in a CYCLE.

\(7^1\) --> units digit of 7.

\(7^2\) --> units digit of 9. (Since the product of the preceding units digit and 7 = 7*7 = 49.)

\(7^3\) --> units digit of 3. (Since the product of the preceding units digit and 7 = 9*7 = 63.)

\(7^4\) --> units digit of 1. (Since the product of the preceding units digit and 7 = 3*7 = 21.)

From here, the units digits will repeat in the same pattern: 7, 9, 3, 1.

The units digit repeat in a CYCLE OF 4.

Implication:

When an integer with a units digit of 7 is raised to a power that is a multiple of 4, the units digit will be 1.

Thus, \(7^8\) has a units digit of 1.

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