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If you divide 7^131 by 5, which remainder do you get?

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SidKan
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Re: If you divide 7^131 by 5, which remainder do you get? [#permalink] If you divide 7^131 by 5, which remainder do you get?   08 Apr 2018, 07:00
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whenever you come across a question like this :-- a number raised to higher power of integer divided by 2,3,4,5,6,7,8,9..
and asking for remainder...then, the question is actually asking for the unit digit of the numerator..

now,
3^4 has a unit digit of 1 => unit digit of 1 will repeat after every fourth power of 3.

6 to the power anything, always has a unit digit of 6

7^4 also has a unit digit of 1 => unit digit of 1 will repeat after every fourth power of 7

now, 151 = 148 + 3

7^148 * 7^3 = unit digit of 1 * unit digit of 3 = unit digit of 3..

any number divided by 5 gives remainder = unit digit of the number, when unit digit is less than 5

and (unit digit - 5) , when unit digit is greator than 5
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Re: If you divide 7^131 by 5, which remainder do you get? [#permalink] If you divide 7^131 by 5, which remainder do you get?   09 Apr 2018, 16:31
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blackcrow wrote:
If you divide 7^131 by 5, which remainder do you get?

A. 0
B. 1
C. 2
D. 3
E. 4



To determine the remainder when 7^131 is divided by 5, we need only to divide the units digit of 7^131 by 5.

Let’s determine the units digit of 7^131:

When writing out the pattern of 7 raised to positive integer exponents, notice that we are concerned ONLY with the units digit of 7 raised to each power.

7^1 = 7

7^2 = 9

7^3 = 3

7^4 = 1

7^5 = 7

The pattern of the units digits of powers of 7 repeats every 4 exponents. The pattern is 7–9–3–1. In this pattern, we see that all positive exponents that are multiples of 4 will produce a 1 as its units digit. Thus:

7^132 has a units digit of 1 and so 7^131 has units digit of 3. So, the remainder of 3/5 = 3.

Answer: D
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Re: If you divide 7^131 by 5, which remainder do you get? [#permalink] If you divide 7^131 by 5, which remainder do you get?   15 Aug 2019, 07:52
blackcrow wrote:
If you divide 7^131 by 5, which remainder do you get?

A. 0
B. 1
C. 2
D. 3
E. 4


\(rof: 7^{131}/5\)
\(rof: (5+2)^{131}/5\)
\(rof: (2)^{131}/5\)
cyclicity rof \(2^{n}/5={2/5, 4/5, 8/5, 16/5…} = {2,4,3,1…}\)
\(rof: (2)^{131}/5\) is \(131/4=3rd.position=3\)

Answer (D).
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If you divide 7^131 by 5, which remainder do you get? [#permalink] If you divide 7^131 by 5, which remainder do you get?   09 Sep 2020, 12:37
This is a method i picked up from solving similar questions and was posted by Bunuel and Karishma in another thread.

7^131 = (5+2)^131 = 5^131 + 2^131

5^131 when divided by 5 doesn't leave a remainder.

for 2^131, find cyclicity of 2 which is 4. Then divide 131 by 4 the remainder would be 3

this means 2^131 = 2^3 in the sense both have same units digits which is 8. now 8/5 = 3

This method has so far worked for me for many questions.
Thanks team.
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Re: If you divide 7^131 by 5, which remainder do you get? [#permalink] If you divide 7^131 by 5, which remainder do you get?   27 Oct 2020, 04:16
Cyclicity of 7 is: 7, 9, 3, 1

When we divide 131 by 4 we get a remainder of 3. Therefore, the units digit of 7^131 is 3.

Dividing 3 by 5, we get remainder 3. Answer is D.
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If you divide 7^131 by 5, which remainder do you get? [#permalink] If you divide 7^131 by 5, which remainder do you get?   15 Sep 2021, 08:23
7^131=(5+2)^131

we have to find the remainder of 2^131 when divided by 5.
2^1 yields a remainder of 2 when divided by 5,
2^2 yields 4
2^3 yields 3
2^4 yields 1
2^5 yields 2
so with a pattern of 2^(4k+3) , we will get 3 as a remainder if divided by 5.
ANS: D

is my method correct ?
Bunuel VeritasKarishma
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Re: If you divide 7^131 by 5, which remainder do you get? [#permalink] If you divide 7^131 by 5, which remainder do you get?   15 Sep 2021, 09:58
Asked: If you divide 7^131 by 5, which remainder do you get?

7^131Mod5 = 49^65*7mod5 = (-1)^65*2mod5 = -2mod5 = 3

IMO D
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Re: If you divide 7^131 by 5, which remainder do you get? [#permalink] If you divide 7^131 by 5, which remainder do you get?   15 Sep 2021, 10:17
blackcrow wrote:
If you divide 7^131 by 5, which remainder do you get?

A. 0
B. 1
C. 2
D. 3
E. 4


(7^2)^65 * 2
=> 4^64 *3
=> (16)^32 *3
=> 1^32 *3
=> 3

Therefore IMO D
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Re: If you divide 7^131 by 5, which remainder do you get? [#permalink] If you divide 7^131 by 5, which remainder do you get?   15 Sep 2021, 22:10
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Jihyo wrote:
7^131=(5+2)^131

we have to find the remainder of 2^131 when divided by 5.
2^1 yields a remainder of 2 when divided by 5,
2^2 yields 4
2^3 yields 3
2^4 yields 1
2^5 yields 2
so with a pattern of 2^(4k+3) , we will get 3 as a remainder if divided by 5.
ANS: D

is my method correct ?
Bunuel VeritasKarishma



Yes, it is.
Note that when dividing by 5, the units digit alone determines the remainder. Since cyclicity of units digit of both 7 and 2 is 4, we know without calculating that cyclicity of remainders will be 4 too. Hence we don't even need to use binomial in the first step.

Check these posts:
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2015/1 ... questions/
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2015/1 ... ns-part-2/
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Re: If you divide 7^131 by 5, which remainder do you get? [#permalink] If you divide 7^131 by 5, which remainder do you get?   06 Apr 2024, 13:26
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