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If x and y are positive integers and r is the remainder when (7^(4x+3)

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If x and y are positive integers and r is the remainder when (7^(4x+3) [#permalink] New post   01 Oct 2018, 04:47
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If x and y are positive integers and r is the remainder when \((7^{4x+3} + y)\) is divided by 10, what is the value of r ?

(1) x = 10
(2) y = 2
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Re: If x and y are positive integers and r is the remainder when (7^(4x+3) [#permalink] New post   01 Oct 2018, 04:53
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Bunuel wrote:
If x and y are positive integers and r is the remainder when \((7^{4x+3} + y)\) is divided by 10, what is the value of r ?

(1) x = 10
(2) y = 2


Question: What is the remainder when \((7^{4x+3} + y)\) is divided by 10

CONCEPT: When a number is divided by 10 then the remainder will always be the unit digit of the number e.g. 37 divided by 10 leaves remainder 7 and 125 divided by 10 leaves remainder 5

i.e. we need to calculate the unit digit of \((7^{4x+3} + y)\)

but Unit digit of \((7^{4x+3})\) is always same as unit digit of \(7^3\) because cyclicity of unit digit of 7 is 4 i.e. Unit digit of powers of 7 repeat after every 4 powers

Hence we only need to know the Unit digit of y to answer the question

Statement 1: x = 10

NOT SUFFICIENT

Statement 2: y = 2

SUFFICIENT

Answer: Option B
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Re: If x and y are positive integers and r is the remainder when (7^(4x+3) [#permalink] New post   15 Oct 2019, 12:38
Bunuel wrote:
If x and y are positive integers and r is the remainder when \((7^{4x+3} + y)\) is divided by 10, what is the value of r ?

(1) x = 10
(2) y = 2



Knowing the ciclicity of 7 the problem becomes a cakewalk.

considering consecutive powers of 7 the unit digits will be

9 (7*7)
3 (9*7)
1 (3*7)
7 (1*7)

every time the power of 7 is a multiple of 4 the unit digit will be 1.

The question stems states: 7^(4x+3)
Hence the unit digit of 7 will be always 3.

According to statement 1 we still miss y. Hence insufficient

Statement 2 gives us y and hence it is sufficient


Option B
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If x and y are positive integers and r is the remainder when (7^(4x+3) [#permalink] New post   15 Oct 2019, 14:33
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Bunuel wrote:
If x and y are positive integers and r is the remainder when \((7^{4x+3} + y)\) is divided by 10, what is the value of r ?

(1) x = 10
(2) y = 2


Analyzing the question:
Finding the remainder when dividing by 10, is the same as finding the units digit of \((7^{4x+3} + y)\). Note \(7^4\) ends in a units digit of 1. So \({(7^4)}^x *7^3\), has a fixed and known units digit. Another way to understand this is the powers of 7 repeat their last digits in a cycle of 4, so \(7^3\), \(7^7\), \(7^{4x + 3}\) etc have the same last digit. Then in order to find the last digit of \((7^{4x+3} + y)\), we are only concerned about y.

Statement 1: Insufficient.
Statement 2: Sufficient.

Ans: B
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If x and y are positive integers and r is the remainder when (7^(4x+3) [#permalink]
15 Oct 2019, 14:33
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