Problem 26 · 2021 Math Kangaroo
Stretch
Number Theory
difference-of-squaresfactorizationprimes
Each of the numbers a and b is the square of an integer. The difference \(a-b\) is a prime number. Which of the following could be the number b?
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Answer: D — 900
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Hint 1 of 2
If a and b are squares, a - b factors as (√a-√b)(√a+√b); for a prime the first factor must be 1.
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Hint 2 of 2
So √a and √b are consecutive, and 2√b+1 must be prime — test each b.
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Approach: difference of squares forces consecutive roots
- With a = m² and b = n², a - b = (m-n)(m+n); a prime needs m-n = 1.
- Then a - b = m+n = 2n+1, which must be prime.
- For b = 900, n = 30 gives 2(30)+1 = 61, which is prime (and a = 961 = 31²).
- The other options give composite differences, so b = 900, choice (D).
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