A number theory problem by Julian Yu
Number Theory
Level
3
When 29 is added to a number, it becomes a perfect square. When 100 is added to the same number, it becomes a perfect fourth power. What is the number?
The answer is 1196.
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1 solution
Let x be the number in question such that:
x + 29 = a^2 (i) x + 100 = b^4 (ii)
where a and b are positive integers. Subtracting (i) from (ii) yields 71 = b^4 - a^2 = (b^2 + a)(b^2 - a). Since 71 is prime it only has 1 and 71 for divisors. We now set up the following system of equations in a and b:
b^2 + a = 71; b^2 - a = 1
which produces a = 35 & b = 6 respectively. Hence, our number x is just x = 35^2 - 29 = 6^4 - 100 = 1196.