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Let a, b, c be natural numbers with a > b, satisfying c² - a² - b² = 101 and a*b = 72. Which one of the following statements is INCORRECT?

1. b and c are coprime
2. c is an odd number
3. a + b + c is an odd number
4. a + b = c + 2

Correct answer: a + b = c + 2

Solution

From the two equations we get (a+b+c)(a+b-c) = 43. Since 43 is prime, the only factorisation gives a+b+c = 43 and a+b-c = 1, so a+b = 22 and c = 21. With a > b and ab = 72 and a+b = 22: a = 4, b = 18 fails since a > b; solving: a and b are roots of t² - 22t + 72 = 0 giving t = 4 or 18, so a = 18, b = 4, c = 21. Check each option against these values.

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