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Remainders!

by maus » Mon Jun 06, 2011 10:40 am
Thanks in advance for your help on this problem:

If x, y, z are positive integers, what is the remainder of 100x+10y+z when divided by 7?

1) y = 6
2) z = 3

I'm horrible on remainders - tips on how to look at these problems to make them easier are welcomed :)
Maus
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by Frankenstein » Mon Jun 06, 2011 10:50 am
Hi,
Let the expression 100x+10y+z be represented by 'p'.
From(1): p = 100x+60+z. Depends on values of x and z
Insufficient
From(1): p = 100x+10y+3. Depends on values of x and y
Insufficient
Both(1) and (2):
p=100x+63. Depends on the value of x
Insufficient

Hence E
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by Whitney Garner » Mon Jun 06, 2011 10:55 am
maus wrote:Thanks in advance for your help on this problem:

If x, y, z are positive integers, what is the remainder of 100x+10y+z when divided by 7?

1) y = 6
2) z = 3

I'm horrible on remainders - tips on how to look at these problems to make them easier are welcomed :)
Hi maus!

Unfortunately, remainder theory isn't nearly as helpful here as simply testing a couple of quick values.

Statement (1):
When y=6, we have a 3 digit number with the 10s digit of 6. Because statement (2) is going to tell us information about the units digit, maybe we can try altering the units digit for 2 different 3 digit numbers. For example: 162 and 163. If numbers are 1 apart, the remainders will be different when divided by 7 (but you can test it to check). Insufficient - Eliminate choices A and D.

Statement (2):
When z=3, we have a 3 digit number with the units digit of 3. Now we can do something similar to what we did with Statement (1). How about 163 and 173. If two numbers are exactly 10 apart, their remainders will be different when divided by 7 (because the 10s are not all multiples of 7, but you can test it to check). Insufficient - Eliminate choice B.

Statement (1+2):
With both y=6 and z=3, we have a 3 digit number with the tens digit 6 and the units digit 3. Again we can test 2 different numbers: 163 and 263. If two numbers are exactly 100 apart, their remainders will be different when divided by 7 (because the 100s are not all multiples of 7, but you can test these two to check). Insufficient - Eliminate choice C, the answer is E.

:)
Whit
Last edited by Whitney Garner on Mon Jun 06, 2011 11:03 am, edited 1 time in total.
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by Frankenstein » Mon Jun 06, 2011 11:00 am
Hi Whitney,
You have mistyped D for E. Kindly edit that part.
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by Whitney Garner » Mon Jun 06, 2011 11:03 am
Frankenstein wrote:Hi Whitney,
You have mistyped D for E. Kindly edit that part.
Thank you for the catch! edited :)
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by cans » Mon Jun 06, 2011 5:16 pm
If x, y, z are positive integers, what is the remainder of 100x+10y+z when divided by 7?

1) y = 6
2) z = 3
100x+10y+z
a)y=6. if we take x,y as same and keep on changing z, we will get different remainders Thus insufficient.
b)z=3. keep x same and z as 6. 10y=10,20,30....
10 remainder = 3
20 remainder = 6
Insufficient.
a&b together) 100x+60+3 = 100x+63. 63 is divisible by 7, thus remainder is based on 100x
x=1, 100 remainder=2
x=2, 200 remainder=4
IMO E
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by Jeff@TargetTestPrep » Tue Dec 12, 2017 4:47 pm
maus wrote:Thanks in advance for your help on this problem:

If x, y, z are positive integers, what is the remainder of 100x+10y+z when divided by 7?

1) y = 6
2) z = 3
We need to determine the remainder of (100x + 10y + z)/7. If we can determine the remainder of 100x/7, 10y/7, and z/7, then we can determine the remainder of (100x + 10y + z)/7.

Statement One Alone:

y = 6

Using the information in statement one, we have:

(100x + 60 + z)/7

Although we know the remainder of 60/7 is 4 (note: 60/7 = 8 + 4/7), we still cannot determine the remainder of 100x/7 or z/7. Statement one alone is not sufficient to answer the question.

Statement Two Alone:

z = 3

Using the information in statement two, we have:

(100x + 10y + 3)/7

Although we know the remainder of 3/7 is 3, we still cannot determine the remainder of 10y/7 or 100x/7.

Statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Using the information from statements one and two, we have:

(100x + 60 + 3)/7 = (100x + 63)/7

Although we know the remainder of 63/7 is 0, we still cannot determine the remainder of 100x/7. Different values of x might yield different remainders. For example, if x = 1, then the remainder of 100/7 is 2, since 100/7 = 14 + 2/7. However, if x = 2, then the remainder of 200/7 is 4, since 200/7 = 28 + 4/7.

Answer: E

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by Brent@GMATPrepNow » Wed Dec 13, 2017 3:54 pm
maus wrote: If x, y, z are positive integers, what is the remainder of 100x+10y+z when divided by 7?

1) y = 6
2) z = 3
Target question: What is the remainder of 100x+10y+z when divided by 7

IMPORTANT: When I scan the statements, I see that even when we COMBINE the statements, we still don't have any information about x, the hundreds digit.
As such, I have a feeling that the combined statements are not sufficient.
So, I'm going to go straight to...

Statements 1 and 2 combined
There are several cases that satisfy BOTH statements. Here are two:
Case a: x = 1, y = 6 and z = 3. Here, 100x + 10y + z = 100(1) + 10(6) + 3 = 163. When we divide 163 by 7 we get 23 with remainder 2. In this case, the answer to the target question is 2
Case b: x = 2, y = 6 and z = 3. Here, 100x + 10y + z = 100(2) + 10(6) + 3 = 263. When we divide 263 by 7 we get 37 with remainder 4. In this case, the answer to the target question is 4
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent
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