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

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Bunuel
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What is the remainder when 3^37 is divided by 10 ? [#permalink] What is the remainder when 3^37 is divided by 10 ?   28 Feb 2023, 07:00
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What is the remainder when 3^37 is divided by 10 ?

A. 1
B. 3
C. 6
D. 7
E. 9
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B
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Abhishek009
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What is the remainder when 3^37 is divided by 10 ? [#permalink] What is the remainder when 3^37 is divided by 10 ?   28 Feb 2023, 07:49
Bunuel wrote:
What is the remainder when 3^37 is divided by 10 ?

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

\(\frac{3^{37}}{10}\)

= \(\frac{3^{4*9+1}}{10}\) ; \(\frac{3^{4}}{10} =\) Reminder \(1\)

So, \(\frac{3^{4*9+1}}{10}\) = Rem 3 , as \(3^{36}\) will have reaminder 1 and 3 will leave remainder 3 when divided by 10, Answer must be 3 (B)....
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Re: What is the remainder when 3^37 is divided by 10 ? [#permalink] What is the remainder when 3^37 is divided by 10 ?   26 Feb 2024, 17:38
Bunuel,

can you please explain it? thanks.
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Bunuel
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Re: What is the remainder when 3^37 is divided by 10 ? [#permalink] What is the remainder when 3^37 is divided by 10 ?   27 Feb 2024, 01:15
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Mansouri wrote:
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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Re: What is the remainder when 3^37 is divided by 10 ? [#permalink]
27 Feb 2024, 01:15
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Bunuel
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