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

[Max@Math Revolution]

[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

[GMATGuruNY]

[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� is divided by 100, we need to know the units digit of 7�.
When an integer is raised to consecutive powers, the resulting units digits repeat in a CYCLE.

7¹ --> units digit of 7.
7² --> units digit of 9. (Since the product of the preceding units digit and 7 = 7*7 = 49.)
7³ --> units digit of 3. (Since the product of the preceding units digit and 7 = 9*7 = 63.)
7� --> 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� has a units digit of 1.

The correct answer is A.

[Brent@GMATPrepNow]

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

A. 1
B. 2
C. 3
D. 4
E. 5

Let's examine 7^8 - 1

Why would I do this?
Well, I know that 7^2 + 1 = 50, which is a factor of 100.
So, perhaps it's the case that 7^8 - 1 is divisible by 100, in which case 7^8 will leave a remainder of 1 when divided by 100

7^8 - 1 is a difference of squares.
So, 7^8 - 1 = (7^4 + 1)(7^4 - 1)
= (7^4 + 1)(7^2 + 1)(7^2 - 1)
= (7^4 + 1)(7^2 + 1)(7 + 1)(7 - 1)
= (7^4 + 1)(50)(8)(6)
= (7^4 + 1)(2400)
= (7^4 + 1)(24)(100)

So, we can see that 7^8 - 1 is divisible by 100
7^8 is 1 greater than 7^8 - 1, so we must get a remainder of 1 when 7^8 is divided by 100

Answer: A

Cheers,
Brent

[Max@Math Revolution]

=>

The remainder when 7^8 is divided by 100 is equal to the final two digits of 7^8.
Now, 7^1 = 7, 7^2 = 49, 7^3 = 343, and 7^4 = 2401.
So, the final two digits of 7^n have period 4:
The tens digits are 0 -> 4 -> 4 -> 0
and the units digits are 7 -> 9 -> 3 -> 1.
It follows that the tens and units digits of 7^8 are 0 and 1, respectively.
Therefore, the remainder when 7^8 is divided by 100 is 1.
Therefore, the answer is A.
Answer : A

[Jeff@TargetTestPrep]

[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

Since 7^4 = (7^2)^2 = 49^2 = 2401, we see that when 7^4, or 2401, is divided by 100, the remainder is 1. Since 7^8 = (7^4)^2, the remainder when 7^8 is divided by 100 will be the same as the square of the remainder when 7^4 is divided by 100. Therefore, that remainder is 1^2 = 1.

Answer: A