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M19 Q11

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M19 Q11 [#permalink] New post 05 Aug 2012, 21:41
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If a and b are positive integers, is a^2 + b^2 divisible by 5 ?

(1) 2ab is divisible by 5
(2) a - b is divisible by 5

I have a question regarding this question. I want to verify the property of multiples that is related to divisibility. So

a) Mutliple of N + Mutliple of N = Multiple of N
b) Mutliple of N + Non -Mutliple of N = Non-Multiple of N
c) Non-Mutliple of N +Non- Mutliple of N = Can be both (multiple or non-multiple)

Does this property work for all integers, are there any exceptions?
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Re: M19 Q11 [#permalink] New post 05 Aug 2012, 23:56
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teal wrote:
If a and b are positive integers, is a^2 + b^2 divisible by 5 ?

2ab is divisible by 5
a - b is divisible by 5



I have a question regarding this question. I want to verify the property of multiples that is related to divisibility. So

a) Mutliple of N + Mutliple of N = Multiple of N
b) Mutliple of N + Non -Mutliple of N = Non-Multiple of N
c) Non-Mutliple of N +Non- Mutliple of N = Can be both (multiple or non-multiple)

Does this property work for all integers, are there any exceptions?


There are no exceptions.

If integers a and b are both multiples of some integer k>1 (divisible by k), then their sum and difference will also be a multiple of k (divisible by k):
Example: a=6 and b=9, both divisible by 3 ---> a+b=15 and a-b=-3, again both divisible by 3.

If out of integers a and b one is a multiple of some integer k>1 and another is not, then their sum and difference will NOT be a multiple of k (divisible by k):
Example: a=6, divisible by 3 and b=5, not divisible by 3 ---> a+b=11 and a-b=1, neither is divisible by 3.

If integers a and b both are NOT multiples of some integer k>1 (divisible by k), then their sum and difference may or may not be a multiple of k (divisible by k):
Example: a=5 and b=4, neither is divisible by 3 ---> a+b=9, is divisible by 3 and a-b=1, is not divisible by 3;
OR: a=6 and b=3, neither is divisible by 5 ---> a+b=9 and a-b=3, neither is divisible by 5;
OR: a=2 and b=2, neither is divisible by 4 ---> a+b=4 and a-b=0, both are divisible by 4.

As for the question:
If a and b are positive integers, is a^2+b^2 divisible by 5 ?

(1) 2ab is divisible by 5 --> if a=b=5 then the answer is YES but if a=5 and b=1 then the answer is NO. Not sufficient.

(2) a-b is divisible by 5 --> if a=b=5 then the answer is YES but if a=b=1 then the answer is NO. Not sufficient.

(1)+(2) From (2) a-b is divisible by 5 so (a-b)^2=(a^2+b^2)-2ab is also divisible by 5. Next, since from (1) 2ab is divisible by 5 then a^2+b^2 must also be divisible by 5 in order their sum to be divisible by 5. Sufficient.

Answer: C.
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If a and b are positive integers [#permalink] New post 25 Jan 2013, 05:09
If a and b are positive integers, is a^2 + b2^2 divisible by 5?

1) 2ab is divisible by 5
2) a-b is divisble by 5

How would you solve this question? Would you pick numbers or would you try algebra? Anyone know the algebraic way of solving this question?
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Re: If a and b are positive integers [#permalink] New post 25 Jan 2013, 05:37
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alexpavlos wrote:
If a and b are positive integers, is a^2 + b2^2 divisible by 5?

1) 2ab is divisible by 5
2) a-b is divisble by 5

How would you solve this question? Would you pick numbers or would you try algebra? Anyone know the algebraic way of solving this question?


Merging similar topics. Please ask if anything remains unclear.
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: If a and b are positive integers   [#permalink] 25 Jan 2013, 05:37
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