If x and y are positive integers and n = 5^x + 7^(y + 15), what is the

Question

If x and y are positive integers and  n = 5^x + 7^{(y + 15)} , what is the units digit of n?

(1)  y = 2x – 15

(2)  y^2 – 6y + 5 = 0

WoundedTiger · Accepted Answer

Sol:n = 5^x + 7^(y + 15)

Note that 5^ (any positive Integer) gives unit digit of 5...So we need to know only y to be able solve the problem..

Cylcity of 7 is 4

7^1=7
7^2=4 9 
7^3=34 3 
7^4=XXX1

7^5=XXXX7

St 1 says y=2x-15 or y+15=2x..thus we know power of 7 in the expression is even so unit digit for 7^(y+15) will be =9 or 1..
We have 2 answers possible for unit digit of n i.e 5+9=1 4 (unit digit) or 5+ 1 =6.

Not sufficient

St 2 says gives value of y=1,5 so we have either 7^16 or 7^20...for each expression unit digit is 1..

Sufficient

Ans B

Bunuel · Answer

Check  Units digits, exponents, remainders problems directory  in our  Special Questions Directory .

thefibonacci · Answer

B.

n = 5^x + 7^(y+15) 
5^x always ends in 5. 
so n = 5 + 7^(y+15)

1) y = 2x-15 
=> y+15 = 2x 
so (y+15) is always even 
but 7^2x can end in 9 or 1 
so insufficient.

2) y^2 - 6y + 5 = 0 
y = 1 or 5 
=> y+15 = 16, 20 (which always ends in 1) 
so sufficient.

Edivar · Answer

Hi.

I did this question and I don't understand why the answers ignore the units digit of 5^x.

if x is even, the units of 5^x is 0, but if x is odd, it is 5.
The question is about the units of n, not 7^(y+15), and only with statement (2) you don't have information about x.

Bunuel · Answer

If x is a positive integer, the units digit of  5^x  is always 5: 5^1 = 5, 5^2 = 25, 5^3 = 125, 5^4 = 625, ... I think you are mixing  5^x  (5 to the power of x) with  5x  (5 multiplied by x).

anubhavece · Answer

Can you please tell me, what would be the answer if roots of quadratic equation were (2,1)? I believe, answer should be B.
Even in this new case we can derive value for (n), however, unit digit for 7 will not match (as in the present case when roots are 5,1).

chetan2u · Answer

hi..

In this case the statement II would read  y^2-3y+2=0 
so when y = 1 or 2, the units digit will change..
in  n = 5^x + 7^{(y + 15)} ..
5^x will always give you 5
 7^{(y+15)}  will become 7^(16) or 7^(17)
7^16 = 7^(4*4+0) so will give you same units digit as 7^4 or 1
7^17 will give 7^(4*4+1) so will give same units digit as 7^1 or 7
so ans will be 5+1=6 OR 5+7=12 or 2
thus insuff

anubhavece · Answer

Ok Thanks a lot
My Bad ..... I forgot that the purpose in DS is not just to find a answer but also remove ambiguity.

GMATinsight · Answer

Question  : Unit Digit of n = ?

Given:  n = 5^x + 7^{(y + 15)}

Point to Note :  5^x  will always have unit digit 5 for any positive integer value of x (as given)
Hence, Calculating value of x is completely immaterial for us

All we need is the unit digit of  7^{(y + 15)}  to find the unit digit of n

Statement 1 :  y = 2x – 15

7^{(y + 15)}  becomes  7^{(2x)} 
for x = 1,  7^{(2x)}  will have unit digit = 9
for x = 2,  7^{(2x)}  will have unit digit = 1

NOT SUFFICIENT

Statement 2 :  y^2 – 6y + 5 = 0

i.e. y = 1 or 5

for y = 1,  7^{(y+15)}  will have unit digit = 1
for y = 5,  7^{(y+15)}  will have unit digit = 1

SUFFICIENT

Answer: option B

Salsanousi · Answer

given that x and y are positive integers they are >0

5^1  has a unit digit of 5
 5^2  has a unit digit of 5. This means that regardless of the exponential power of 5 it will always have a digit of 5, as long as the power is a positive integer.

So we are interested in knowing the y value.

Statement 1) does not provide us any information.

Insufficient

Statement 2)  y^2 – 6y + 5 = 0

(y-5)(y-1) = 0
y = 1 or 5

The unit digit of powers of 7 are below.
 7^1 =7
 7^2 = 49 
 7^3 = 343 
 7^4 = 2401

This cycle repeats in multiples of 4, so 8,12,16,20 would have a unit digit of 1.

if y =1 then the power is 16, meaning it will have a units digit of 1
if y = 5 then the power is 20, meaning again it will have a digit of 1.

sufficient.

B is the answer.

Basshead · Answer

If x and y are positive integers and  n = 5^x + 7^{(y + 15)} , what is the units digit of n?

Quick notes:
5 to any power = units digit of 5.
Cyclicity of 7 = 7, 9, 3, 1

To determine the units digit of n, we need to determine the value of y+15. Lets look at the choices:

(1)  y = 2x – 15

We have no idea what x is --  Insufficient .

(2)  y^2 – 6y + 5 = 0

y = 5, 1

y + 15 = 16 or 20.

Since y has a cyclicity of 4, y raised to any multiple of 4 will give us a units digit of 1.

Statement 2 is sufficient.

Answer is B.

nivivacious · Answer

Trick here is not to fall into C trap
5^anything = 5
Hence finding x is not important

y^2 – 6y + 5 = 0
We get solutions as 5 and 1
for both, the unit digit ie 7^20 and 7^16 is the same ie 9
Hence B

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

If x and y are positive integers and n = 5^x + 7^(y + 15), what is the

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

If x and y are positive integers and n = 5^x + 7^(y + 15), what is the

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards