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# If a and b are positive integers, what is the remainder when 74a+b is

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Senior SC Moderator
Joined: 14 Nov 2016
Posts: 1305
Location: Malaysia
If a and b are positive integers, what is the remainder when 74a+b is [#permalink]

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24 Mar 2017, 05:40
00:00

Difficulty:

45% (medium)

Question Stats:

68% (01:00) correct 32% (01:22) wrong based on 60 sessions

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If a and b are positive integers, what is the remainder when $$7^{4a}+b$$ is divided by 4?

1) a=2
2) b=3

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Senior Manager
Joined: 13 Oct 2016
Posts: 367
GPA: 3.98
Re: If a and b are positive integers, what is the remainder when 74a+b is [#permalink]

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24 Mar 2017, 08:36
1
ziyuen wrote:
If a and b are positive integers, what is the remainder when $$7^{4a}+b$$ is divided by 4?

1) a=2
2) b=3

Hi

$$7^{4a} + b$$

$$7^2$$ leaves remainder $$1$$ when divided by $$4$$ ------> $$(7^2)^{2a} + b = 1^{2a} + b = 1 + b$$

To solve the question we need to know only the remainder that $$b$$ leaves while divided by $$4$$.

Manager
Joined: 08 Sep 2015
Posts: 70
Re: If a and b are positive integers, what is the remainder when 74a+b is [#permalink]

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24 Mar 2017, 10:07
To solve this question, have to remember the last digit of a power...for 7 the cyclicity is 7,9,3,1 -> since a is a positive integer, the 7^4a will always end with 1 and we know that a number is divisible by 4 when the last two digits are divisible by 4.

A) Irrelevant -> not sufficient
B) since we know b, we know the last digit of our number, therefore can find the remainder. Sufficient.
Re: If a and b are positive integers, what is the remainder when 74a+b is   [#permalink] 24 Mar 2017, 10:07
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