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

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If a and b are positive integers, what is the remainder when [#permalink]  26 Feb 2012, 15:30
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If a and b are positive integers, what is the remainder when $$4^{2a+1+b}$$ is divided by 10?

(1) a = 1
(2) b = 2

[Reveal] Spoiler:
Ok - this is how I am trying to solve this.

Statement 1

a = 1. Does not tell anything about b --therefore is insufficient on its own to answer the question.

Statement 2

b = 2

2a + 1 + b becomes

2a (even) + 1 (odd) + b (even) = ODD. So the exponent to 4 is ODD. So I understand that if we put 3, 5 etc I get the remainder 4, but why can't I put exponent as 1 as 1 is ODD too. Can you please help?
[Reveal] Spoiler: OA

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Re: Remainder when divided by 10 [#permalink]  26 Feb 2012, 15:39
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powers of 4 go like this:

The unit place is:
1 = 4
2 = 6
3 = 4
4 = 6

So all even exponents have 6 in unit place, and all off exponents have 4 in unit place. To solve the problem we need to find whether the 2a + 1 + b is even or odd

As a is +ve integer, 2a is always even. 2a + 1 will be odd. Now to determine whether (2a + 1 + b) is even or odd, we need to know only b.

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Re: Remainder when divided by 10 [#permalink]  26 Feb 2012, 15:58
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If a and b are positive integers, what is the remainder when $$4^{2a+1+b}$$ is divided by 10?

This is a classic "C trap" question: "C trap" is a problem which is VERY OBVIOUSLY sufficient if both statements are taken together. When you see such question you should be extremely cautious when choosing C for an answer.

Back to the question: 4 in positive integer power can have only 2 last digits: 4, when the power is odd or 6 when the power is even. Hence, to get the remainder of 4^x/10 we should know whether the power is odd or even: if it's odd the remainder will be 4 and if it's even the remainder will be 6.

(1) a = 1 --> $$4^{2a+1+b}=4^{3+b}$$ depending on b the power can be even or odd. Not sufficient.

(2) b = 2 --> $$4^{2a+1+b}=4^{2a+3}=4^{even+odd}=4^{odd}$$ --> the remainder upon division of $$4^{odd}$$ by 10 is 4. Sufficient.

enigma123 wrote:
2a (even) + 1 (odd) + b (even) = ODD. So the exponent to 4 is ODD. So I understand that if we put 3, 5 etc I get the remainder 4, but why can't I put exponent as 1 as 1 is ODD too. Can you please help?

The power of 4 is $$2a+3$$ and since $$a$$ is a positive integer then the lowest value of $$2a+3$$ is 5, for $$a=1$$. Next, even if the power were 1 then 4^1=4 and the remainder upon division of 4 by 10 would still be 4.

Hope it's clear.
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Re: If a and b are positive integers, what is the remainder when [#permalink]  11 Aug 2013, 08:20
REM(4^(2a+1+b))/10

Means we have to find last digit of the expression.So rephrasing the question

What is the last digit of 4^(2a+1+b)

(1).
a=1

Break the expression as 4^2a * 4^1 * 4^b.

b is unknown hence INSUFFICIENT

(2).

b=2

4^2a * 4^1 * 4^b.

If you can observe the expression 4^2a, you will see that this will always give last digit as '6' you can try out numbers if you want.

So knowing the expression and value of b last digit can be calculated and hence the remainder can also be calculated.

Hence (B) it is !!
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If a and b are positive integers, what is the remainder when [#permalink]  24 Aug 2013, 07:39
If a and b are positive integers, what is the remainder when 412a+3+9b is divided by 10?
(1) a = 1
(2) b = 2
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Re: If a and b are positive integers, what is the remainder when [#permalink]  24 Aug 2013, 07:54
AMITAGARWAL2 wrote:
If a and b are positive integers, what is the remainder when 412a+3+9b is divided by 10?
(1) a = 1
(2) b = 2

remainder when 412a+3+9b is divided by 10 = 2a + 3 + 9b

Statement 1
a=1
We don't know the value of "b".
Thus Insufficient

Statement 2
b=2
We don't know the value of "a".
Thus Insufficient

Statement 1 & 2
a=1 & b=2
Thus Sufficient

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Re: If a and b are positive integers, what is the remainder when [#permalink]  24 Aug 2013, 13:50
fameatop wrote:
AMITAGARWAL2 wrote:
If a and b are positive integers, what is the remainder when 412a+3+9b is divided by 10?
(1) a = 1
(2) b = 2

remainder when 412a+3+9b is divided by 10 = 2a + 3 + 9b

Statement 1
a=1
We don't know the value of "b".
Thus Insufficient

Statement 2
b=2
We don't know the value of "a".
Thus Insufficient

Statement 1 & 2
a=1 & b=2
Thus Sufficient

Me too got c as the answer. Can you pls explain how b is the oA?
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Re: If a and b are positive integers, what is the remainder when [#permalink]  25 Aug 2013, 06:51
Expert's post
AMITAGARWAL2 wrote:
If a and b are positive integers, what is the remainder when 412a+3+9b is divided by 10?
(1) a = 1
(2) b = 2

Merging similar topics.

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Re: If a and b are positive integers, what is the remainder when [#permalink]  30 Aug 2014, 04:22
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Re: If a and b are positive integers, what is the remainder when   [#permalink] 30 Aug 2014, 04:22
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