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Math Expert V
Joined: 02 Sep 2009
Posts: 58453
If x and y are positive integers and r is the remainder when (7^(4x+3)  [#permalink]

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Difficulty:   45% (medium)

Question Stats: 55% (01:11) correct 45% (01:30) wrong based on 60 sessions

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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]

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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

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Re: If x and y are positive integers and r is the remainder when (7^(4x+3)  [#permalink]

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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

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
GMAT Tutor S
Joined: 17 Sep 2014
Posts: 198
Location: United States
GMAT 1: 780 Q51 V45 GRE 1: Q170 V167 If x and y are positive integers and r is the remainder when (7^(4x+3)  [#permalink]

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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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