For how many unique pairs of nonnegative integers {a, b} is the equati

Question

For how many unique pairs of nonnegative integers {a, b} is the equation a^2 - b^2 = 225 true?

A) 1
B) 3
C) 5
D) 7
E) 9

Kudos for a correct  solution .

Bunuel · Accepted Answer

MANHATTAN GMAT OFFICIAL SOLUTION: If a^2-b^2=225, then (a+b)(a-b)=225. The problem specifies that a and b are nonnegative, so a + b must represent the larger of the two factors and a – b must represent the smaller of the two factors. This is true in all cases but one. What is that one case? Quick, what’s the square root of 225? If you don’t know that it’s 15, add this to your to-memorize list. The two factors could both equal 15; in this case, a = 15 and b = 0. What are the other possible factors of 225? Break the 15s down to primes: 3 ´ 5 ´ 3 ´ 5, or 3^2´ 5^2. If a number equals 3^2´ 5^2, then the factors could be made of any combination of 3^0, 3^1, 3^2 and 5^0, 5^1, and 5^2. Because there are three options for each of two base prime numbers, there are 3 ´ 3 = 9 distinct factors for this number. Find the lowest numbers first and then do the math to find the factor pair: 3^0*5^0=1. This must be paired with 225 (because 1 × 225 = 225). 3^1*5^0=3. This must be paired with 225 / 3 = 75. 3^0*5^1=5. This must be paired with 225 / 5 = 45. 3^2*5^0=9. This must be paired with 225 / 9 = 25. 3^1*5^1=15. This must be paired with 225 / 15 = 15. That’s a total of 9 factors, so you know you’re done. How can you use that info to find the unique pairs of a and b? Recall that (a+b)(a-b)=225. Take the first pair: Screen-shot-2013-08-19-at-9.35.21-AM.png Given that a+b=225 and that a-b=1, how can you solve for a and b? Subtract equations: a+b=225 -(a-b)=1 _____________________ 2b=(225-1) b=(225-1)/2 In other words, the value of b is equal to (Factor 1 – Factor 2) / 2. By the same token, the value of a is equal to (Factor 1 + Factor 2) / 2. (If you’re not convinced about that, prove it using the same kind of math shown for b above.) So, for factors 225 and 1, a={(225+1)/2}=113 and b={(225-1)/2}=112. Test the remaining factor pairs: Screen-shot-2013-08-19-at-9.36.38-AM.png ** use a calculator! ** (you wouldn’t be required to do this kind of math on the real test) There are, therefore, five unique pairs of {a, b} values for which the equation a^2-b^2=225 is true. The correct answer is (C).

vijay556 · Answer

(a+b)(a-b)=225

225 can be written as 5*5*3*3*1

so 225 can be written as product of two factors in 5 ways
5*45
25*9
225*1
15*15
75*3

solving above three equations, we get Five non negative pairs of (a,b)

Hence C

amgb · Answer

225,1---1 
15,0--2
45,9--3
75,3--4
25,9--5

hence ans c

shriramvelamuri · Answer

my take option A. 15,0. Have to wait another couple of days for OA.

Lucky2783 · Answer

Answer C
(a+b)(a-b)=225

5cases For (a+b), (a-b)

225, 1
75, 3
45, 5
25,20
15, 15

Answer 5

Zhenek · Answer

225 = 1*3*5*3*5 = (a+b)*(a-b)
Its worth mentioning that if a - b > a + b, then b < 0 which doesn't work so we need to pay attention to that
Options:
a+b = 3*3*5*5, a-b = 1
a+b = 3*3*5, a-b = 5
a+b = 3*5*5, a-b = 3
a+b = 3*5, a-b = 3*5
a+b = 5*5, a-b = 3*3
rest of the options don't work coz a-b ends up bigger than a+b

C is the answer

mvictor · Answer

almost got me tricked this question...
ok, so a^2-b^2=225
we can re-write it as:
(a+b)(a-b)=225 or 15^2.
15^2 can be rewritten as 3^2 * 5^2. we thus have 9 factors.
let's see:
225; 1
75; 3
45; 5
9; 25
15; 15

since all are negative, for each "pair" we can find unique values for a and b to satisfy the conditions:
a+b=225; a-b=1
a+b=75; a-b=3
a+b=45; a-b=5
a+b=25; a-b=9
a+b=15; a-b=15

since we are asked only non-negative numbers, we can suppose that one of the numbers is 0.
thus, we see that we have 5 such pairs.
but all this is too time consuming, is there another approach?

kajaldaryani46 · Answer

——
So I believe in this Question, finding out the factor pairs using prime factorization is enough. I was stuck and kept trying to find the exact numbers a and b such that the product of their sum and their difference would be equal to 225.

I could find 3 pairs ie (15,0), (25,20) and (17,8). But beyond that I couldn’t find those - now I figure that those numbers are quite big and therefore they won’t occur to us naturally. therefore, if we are able to list down the factor pairs, that should be enough

Posted from my mobile device

Archit3110 · Answer

for 225 ; total possible factors; 5^2*3^2 ; 3*3 ; 9
possible pairs
225*1
75*3
45*5
25*9
15*15
total 5 pairs possible

GMATmronevsall · Answer

Easy one.
(a-b)*(a+b)=225
divide 225 by checking the divisibility test.
Number of factors: 5^2*3^2= (2+1)(2+1)=9 
Leaving 1 alone there are 9 factors so 
(9+1)/2 = 5

Answer C

Ayushi1597 · Answer

Hi, 
Could you explain how you to got to "so 225 can be written as product of two factors in 5 ways"

Kinshook · Answer

For how many unique pairs of nonnegative integers {a, b} is the equation a^2 - b^2 = 225 true?

225 = 3^2*5^2

(a+b)(a-b) = 225 = 225*1 = 45*5 = 75*3 = 25*9 = 15*15

Case 1: a+b=225; a-b=1; a= 113; b=112
Case 2: a+b=45; a-b=5; a= 25; b=20
Case 3: a+b=75; a-b=3; a= 39; b=36
Case 4: a+b=25; a-b=9; a= 17; b=8
Case 5: a+b = 15; a-b =15; a= 15; b=0

IMO C

For how many unique pairs of nonnegative integers {a, b} is the equati

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For how many unique pairs of nonnegative integers {a, b} is the equati

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