What is the remainder when 7^548 is divided by 10?

Question

What is the remainder when 7^{548} is divided by 10? A. 1 B. 3 C. 7 D. 8 E. 9 2024-01-25_13-51-58.png

gmatophobia · Accepted Answer

The remainder when  7^{548}  is divided by 10 is the unit digit of  7^{548}

Cyclicity of 7 ⇒ 7, 9, 3, 1

548 is divisible by 4, as 48 is divisible by 4. Hence, we can represent 548 as 4*n

7 ^{548} = (7^{4})^n = 1^n = 1

Option A

Bunuel · Answer

When dividing a positive integer by 10, the remainder is always the units digit of that integer. For instance, 123 divided by 10 yields the remainder of 3. Hence, essentially we need to find the units digit of  7^{548} .

7^{548} = (7^2)^{274} = 49^{274} = (50 - 1)^{274} .

When expanding  (50 - 1)^{274} , all terms but the last one will have 50 as their factors making them divisible by 10 and the last term will be  (-1)^{274}=1 , which when divided by 10 yields the remainder of 1. Here, we could also note that when expanding we'll get all the terms with 50 and the last term  (-1)^{274}=1 , hence the sum would be something with the units digit of 1, giving the remainder of 1 when divided by 10.

Alternatively, we can use the cyclicity of 7 in positive integer power, which is four, meaning that the units digit of 7 in positive integer power repeats in blocks of four {7, 9, 3, 1}{7, 9, 3, 1}... The power, 548, is a multiple of 4, so the units digit of  7^{548}  will be the fourth number in the cyclicity block, which is 1, giving the remainder of 1 when divided by 10.

Answer: A.

Danou · Answer

I am still having quite some trouble understanding how to come to the answer. Is there another approach to look at this?

Bunuel · Answer

Maybe practicing similar questions could help:  Units digits, exponents, remainders problems

VivekSri · Answer

ScottTargetTestPrep · Answer

First we must remember that the remainder when dividing a number by 10 equals the units digit.

So, to determine the unit digit of 7^548, we must first see that 7 raised to a power has units digits in a repeating pattern of 4. If you don’t have this memorized, we can confirm it below:

7^1 has a untis digit of 7

7^2 has a units digit of 9

7^3 has a units digit of 3

7^4 has a units digit of 1

7^5 has a units digit of 7.

So, the pattern of units digits is 7-9-3-1 and most importantly, when 7 is raised to a power that is a multiple of 4, it has a units digit of 1.

Thus, since 548 is a multiple of 4, 7^548 has a units digit of 1, and therefore 5^548 divided by 10 also has a remainder of 1.