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# If x is a positive integer, what is the remainder when 7^(12

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Re: If x is a positive integer, what is the remainder when 7^(12 [#permalink]

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21 Mar 2014, 05:56
jlgdr wrote:
jlgdr wrote:
hogann wrote:
If x is a positive integer, what is the remainder when 7^(12x+3) + 3 is divided by 5?

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

Binomial method works well here.
First lets assume x=1

so 7^12x+3 + 3 / 5 what is the remainder?

7^15 + 3 /5

(5+2)^15 / 5, all terms will be divisible by 5 except 2^15

What is remainder of 2^15 / 5?

2^15 = 2* 4^7 = 2(5-1)^7

All terms will again be divisible by 5 except -1^7
Then -1*2 = -2 + 3 = 1

Hence we have a remainder of 1 when divided by 7

Cheers!
J

Just wondering is it possible to multiply both numerator and denominator by 2 in order to get 10 as denominator? That way we would only need to worry about the units digit of the expression

Bunuel, could you clarify whether this approach is valid?

Thanks
Cheers
J

Not following you.

Aren't we even without multiplying concerned only about the units digit?
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Re: If x is a positive integer, what is the remainder when 7^(12 [#permalink]

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21 Mar 2014, 08:58
Well if we are dividing by 5 then the remainder has nothing to do with the units digit as far as I'm concerned. But if we had 10 as a denominator then we could just care about the units digit on the numerator and find the answer easily, that's what I tried to do but somehow I'm getting 2 as a remainder which of course is not correct
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Re: If x is a positive integer, what is the remainder when 7^(12 [#permalink]

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14 Aug 2015, 10:33
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Re: If x is a positive integer, what is the remainder when 7^(12 [#permalink]

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15 Aug 2015, 00:12
hogann wrote:
If x is a positive integer, what is the remainder when 7^(12x+3) + 3 is divided by 5?

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

7 cong 2 mod 5
7^2 cong 4 mod 5
7^3 cong 3 mod 5
7^4 cong 1 mod 5
7^5 cong 2 mod 5 and so on
the cycle is 2, 4, 3, 1
put x=1,2,3
we get the exponent of 7 as 15, 27 and so on
when you divide 15 or 27 or 39 by 4 you always get a remainder 3
therefore, for the first part of term, the remainder is 3
3+3=6; when divided by 5, the remainder is 1
Hence, the correct option is B
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Re: If x is a positive integer, what is the remainder when 7^(12 [#permalink]

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23 Sep 2016, 00:53
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Re: If x is a positive integer, what is the remainder when 7^(12 [#permalink]

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05 Dec 2016, 01:41
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hogann wrote:
If x is a positive integer, what is the remainder when 7^(12x+3) + 3 is divided by 5?

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

We can solve this question without plugging in numbers.

$$7^{(12x+3)} = 7^{3*(4x+1)}$$

Unit digit of 7 has a cycle of 4 (7, 9, 3, 1)

$$\frac{3*(4x+1)}{4} = \frac{3*(0+1)}{4} = \frac{3}{4}$$ ---> remainder is 3 so our power of 7 is 3 with units digit 3.

and we have $$\frac{3 + 3}{5} = \frac{6}{5}$$

Remainder 1.
Re: If x is a positive integer, what is the remainder when 7^(12   [#permalink] 05 Dec 2016, 01:41

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