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Mansouri
What is the remainder when 3^37 is divided by 10 ?

A. 1
B. 3
C. 6
D. 7
E. 9

Bunuel,

can you please explain it? thanks.
­

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 3^37.

For that, we can use the cyclicity of 3 in positive integer power, which is four, meaning that the units digit of 3 in positive integer power repeats in blocks of four {3, 9, 7, 1}{3, 9, 7, 1}...

    3^1 = 3
    3^2 = 9
    3^3 = 27
    3^4 = 81
    3^5 = 243
    ...
 
The power, 37, is 1 greater than a multiple of 4, so the units digit of 3^37 will be the first number in the cyclicity block, which is 3, giving the remainder of 3 when divided by 10.

Answer: B.­

Theory is here: https://gmatclub.com/forum/math-number- ... 88376.html

Check Units digits, exponents, remainders problems directory in our Special Questions Directory.

Hope it helps.­­­
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What is the remainder of \( 3^{37} \) when divided by 10

Theory: Remainder of a number by 10 is same as the unit's digit of the number

(Watch this Video to Learn How to find Remainders of Numbers by 10)

Using Above theory Remainder of \( 3^{37} \) by 10 = unit's digit of \( 3^{37} \)

Now to find the unit's digit of \( 3^{37} \) , we need to find the pattern / cycle of unit's digit of power of 3 and then generalizing it.

Unit's digit of \(3^1\) = 3
Unit's digit of \(3^2\) = 9
Unit's digit of \(3^3\) = 7
Unit's digit of \(3^4\) = 1
Unit's digit of \(3^5\) = 3

So, unit's digit of power of 3 repeats after every \(4^{th}\) number.
=> We need to divided 37 by 4 and check what is the remainder
=> 37 divided by 4 gives 1 remainder

=> \( 3^{37} \) will have the same unit's digit as \(3^1\) = 3

So, Answer will be B
Hope it helps!

MASTER How to Find Remainders with 2, 3, 5, 9, 10 and Binomial Theorem

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