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If x and y are positive integers and n = 5^x + 7^(y + 3), what is the

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Joined: 02 Sep 2009
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If x and y are positive integers and n = 5^x + 7^(y + 3), what is the  [#permalink]

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12 Jun 2015, 03:15
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If x and y are positive integers and n = 5^x + 7^(y + 3), what is the units digit of n?

(1) y = 2x – 16
(2) y is divisible by 4.

Kudos for a correct solution.

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Re: If x and y are positive integers and n = 5^x + 7^(y + 3), what is the  [#permalink]

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12 Jun 2015, 03:29
1
x is positive ,Hence 5^x will always have units digit 5 and y is positive but Y=2x-16 is not sufficient. Units digit of 7^7 and units digit of 7^5 are different (Here Y=2, y=4) Hence OA=B
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Re: If x and y are positive integers and n = 5^x + 7^(y + 3), what is the  [#permalink]

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12 Jun 2015, 04:01
If x and y are positive integers and n = 5^x + 7^(y + 3), what is the units digit of n?

(1) y = 2x – 16
(2) y is divisible by 4.

Note that 5^x always ends in 5. So we need to know only y to solve the problem.

7^1=7
7^3=343

7^5=16807 ...Every 4 values, units digit repeats

Stmt 1 - y=2x–16 or y+3=2x-13. Thus power of 7 in the expression is odd so unit digit for 7^(y+3) will be = 7 or 3.
We have 2 answers possible for unit digit of n i.e 5+7=12 (unit digit) or 5+3=8.

Not sufficient

Stmt 2 - Value of y=4,8... so we have 7^7 or 7^11...for each expression unit digit is 3.

Sufficient

Ans B
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Re: If x and y are positive integers and n = 5^x + 7^(y + 3), what is the  [#permalink]

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15 Jun 2015, 05:34
Bunuel wrote:
If x and y are positive integers and n = 5^x + 7^(y + 3), what is the units digit of n?

(1) y = 2x – 16
(2) y is divisible by 4.

Kudos for a correct solution.

MANHATTAN GMAT OFFICIAL SOLUTION:

The units digit of n is determined solely by the units digit of the expressions 5^x and 7^(y + 3), because when two numbers are added together, the units digit of the sum is determined solely by the units digits of the added numbers.

Since x is a positive integer, and 5^(any positive integer) always has a units digit of 5, 5^x always ends in a 5.

However, the units digit of 7^(y + 3) is not certain, as the units digit pattern for the powers of 7 is a four-term repeat: [7, 9, 3, 1].

The question can thus be rephrased as “what is the units digit of 7^(y + 3)?”

Note: Determining y would be one way of answering the question above, but we should not rephrase to “what is y?” Because the units digits of the powers of 7 have a repeating pattern, we might get a single answer for the units digit of 7^(y + 3) despite having multiple values for y.

(1) INSUFFICIENT: This statement tells us neither the value of y nor the units digit of 7^(y + 3), as y depends on the value of x, which could be any positive integer. For example, if x = 9, then y = 2 and 7^(y + 3) has a units digit of 7. By contrast, if x = 10, then y = 4 and 7^(y + 3) has a units digit of 3.

(2) SUFFICIENT: Regardless of what multiple of 4 we pick, 7^(y + 3) will have the same units digit. Ultimately this means that n has a units digit of 5 + 3 = 8.

The correct answer is B.
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Re: If x and y are positive integers and n = 5^x + 7^(y + 3), what is the  [#permalink]

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16 Jun 2015, 08:09
Bunuel wrote:
If x and y are positive integers and n = 5^x + 7^(y + 3), what is the units digit of n?

(1) y = 2x – 16
(2) y is divisible by 4.

Kudos for a correct solution.

Analysis :
X > 0 , Y > 0
5 raise to anything (greater than zero) will have 5 in unit digit. 7 has power cycle of 4 - 7,9,3,1
So unit digit will depend on unit digit of n will depend on 7^(y + 3), that means Unit digit of n will depend on Y.

statement 1 :
y = 2x – 16, y can be anything hence statement 1 is not sufficient.

Statement 2 :
y is divisible by 4,

Y = 4K, and Y + 3 = 4K +3

so 7 ^(Y + 3) will have unit digit of 3.

Statement 2 is sufficient.

Hence Option B is correct.
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Re: If x and y are positive integers and n = 5^x + 7^(y + 3), what is the  [#permalink]

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28 Mar 2019, 07:05
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Re: If x and y are positive integers and n = 5^x + 7^(y + 3), what is the   [#permalink] 28 Mar 2019, 07:05
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