Hello,
Can you please assist with this:
If r is the remainder of (2^(8x + 3) + y )/5 and x is
a positive integer, what is the value of r?
(1) y is even
(2) y = 2
OA: B
When I plug in x = 1 I get 2^(8x1 + 3) = 2^11 = 2048
1) Let y = 2.
So (2048 + 2)/5 => Remainder = 0
Let y = 4
So (2048 + 4)/5 => Remainder = 2
Insufficient
2) y = 2
Sufficient
I was wondering if this approach is correct or if there is a easier way so that we can avoid calculating 2^11 ?
Thanks a lot,
Sri
What is the remainder?
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Hi Sri,
This is kind of question where we need to rephrase the given pattern.
You have to note the pattern here,
2^1=2
2^2=4
2^3=8
2^4=_6
after here pattern repeats like 2,4,8,6,2,4,8,6.......
2^8x always ends with 6(since x is positive integer)
so 2^(8x+3) will always end with 8
now question can be re-written as
what is the reminder when (8+Y)/5 ?
now we need value of Y.
Statement 1: Y is even.
when Y =2
8+2/5 =0
When Y =4
8+4/5 =2
Hence Insufficient.
Statement 2: Y = 2
so 8+2/5 = 0.
Hence Sufficient.
Ans is B
Regards,
Uva.
This is kind of question where we need to rephrase the given pattern.
You have to note the pattern here,
2^1=2
2^2=4
2^3=8
2^4=_6
after here pattern repeats like 2,4,8,6,2,4,8,6.......
2^8x always ends with 6(since x is positive integer)
so 2^(8x+3) will always end with 8
now question can be re-written as
what is the reminder when (8+Y)/5 ?
now we need value of Y.
Statement 1: Y is even.
when Y =2
8+2/5 =0
When Y =4
8+4/5 =2
Hence Insufficient.
Statement 2: Y = 2
so 8+2/5 = 0.
Hence Sufficient.
Ans is B
Regards,
Uva.
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Hi Sri!gmattesttaker2 wrote:Hello,
Can you please assist with this:
I was wondering if this approach is correct or if there is a easier way so that we can avoid calculating 2^11 ?
Thanks a lot,
Sri
For remainder questions, we often only need to look at the last 1 or 2 digits of the number. In this case, since we're dividing by 5, only the last digit actually matters.
So, we can recognize the pattern for powers of 2:
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = ..6
2^5 = ..2
and so on. So, the units digit of powers of 2 is a 4 part cycle: 2, 4, 8, 6, ...
Now we just see where 11 falls in the cycle! 11 is one short of 12, which would end the cycle, so 2^11 must end in "8".
As an aside, you didn't actually prove that (2) was sufficient, since you only tested x=1; x could be any positive integer! You should test at least 1 more value of x to be sure.
For x=2:
(2^(19) + y)/5
Hey, 19 is also 2nd last in the cycle and 2^19 has a units digit of 8.
If we're not convinced yet:
For x=3:
(2^(27) + y)/5
and 27 is also 2nd last in the cycle with a units digit of 8.
How could we figure that out without picking numbers? The "8x" in the "8x+3" will always be a multiple of 4. So, 2^8x would always be the last part of the cycle! Since we're adding 3, (2^(8x+3)) will always be 3rd in the cycle and have a units digit of 8.
I hope that helps!
Stuart
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r = (2^(8x + 3) + y )/5
= (2^8x * 8 + y) / 5
= (2^4 * 8 + y)/5 (using remainder rule = 8x/4 = 0.. So, 2^4 will give unit digit)
= ( 128 + y)/5 =
Statement 1:
y = EVEN
y = 0, r = 3
y = 2, r = 0
INSUFFICIENT
Statement 2:
y = 2
130/5 = r = 0
SUFFICIENT
Answer [spoiler]{B}[/spoiler]
= (2^8x * 8 + y) / 5
= (2^4 * 8 + y)/5 (using remainder rule = 8x/4 = 0.. So, 2^4 will give unit digit)
= ( 128 + y)/5 =
Statement 1:
y = EVEN
y = 0, r = 3
y = 2, r = 0
INSUFFICIENT
Statement 2:
y = 2
130/5 = r = 0
SUFFICIENT
Answer [spoiler]{B}[/spoiler]
R A H U L
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Hello Uva,Uva@90 wrote:Hi Sri,
This is kind of question where we need to rephrase the given pattern.
You have to note the pattern here,
2^1=2
2^2=4
2^3=8
2^4=_6
after here pattern repeats like 2,4,8,6,2,4,8,6.......
2^8x always ends with 6(since x is positive integer)
so 2^(8x+3) will always end with 8
now question can be re-written as
what is the reminder when (8+Y)/5 ?
now we need value of Y.
Statement 1: Y is even.
when Y =2
8+2/5 =0
When Y =4
8+4/5 =2
Hence Insufficient.
Statement 2: Y = 2
so 8+2/5 = 0.
Hence Sufficient.
Ans is B
Regards,
Uva.
Thank you very much for your detailed and thorough explanation.
Best Regards,
Sri
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Stuart Kovinsky wrote:Hi Sri!gmattesttaker2 wrote:Hello,
Can you please assist with this:
I was wondering if this approach is correct or if there is a easier way so that we can avoid calculating 2^11 ?
Thanks a lot,
Sri
For remainder questions, we often only need to look at the last 1 or 2 digits of the number. In this case, since we're dividing by 5, only the last digit actually matters.
So, we can recognize the pattern for powers of 2:
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = ..6
2^5 = ..2
and so on. So, the units digit of powers of 2 is a 4 part cycle: 2, 4, 8, 6, ...
Now we just see where 11 falls in the cycle! 11 is one short of 12, which would end the cycle, so 2^11 must end in "8".
As an aside, you didn't actually prove that (2) was sufficient, since you only tested x=1; x could be any positive integer! You should test at least 1 more value of x to be sure.
For x=2:
(2^(19) + y)/5
Hey, 19 is also 2nd last in the cycle and 2^19 has a units digit of 8.
If we're not convinced yet:
For x=3:
(2^(27) + y)/5
and 27 is also 2nd last in the cycle with a units digit of 8.
How could we figure that out without picking numbers? The "8x" in the "8x+3" will always be a multiple of 4. So, 2^8x would always be the last part of the cycle! Since we're adding 3, (2^(8x+3)) will always be 3rd in the cycle and have a units digit of 8.
I hope that helps!
Stuart
Hello Stuart,
Thank you very much for your detailed and excellent explanation (as always).
Best Regards,
Sri