Bunuel wrote:
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.
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:
Attachment:
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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
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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:
Attachment:
Screen-shot-2013-08-19-at-9.36.38-AM.png [ 13.65 KiB | Viewed 5069 times ]
** 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).
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