Nested Cube Roots
If 3 2 7 8 0 + 3 2 7 8 0 + 3 2 7 8 0 + . . . = b a where a and b are co-prime numbers.
What is the value of a + b ?
The answer is 8.
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1 solution
How do you know that a(a+b)(a-b) does not have common factors with b^3
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GCD(a,b)=GCD(a±b,b) for all integers.
(This is a fairly fundamental rule, which Wikipedia lists under the properties of the GCD as "If m is any integer, then gcd(a + m·b, b) = gcd(a, b)." It's exploited, for example, in finding the GCD of two numbers through the Euclidean Algorithm)
Let a/b=x, so x³-80/27=x
x(x²-1)=80/27
a/b (a+b)/b (a-b)/b=80/27
(Since you said a/b is simplified, and 27=3³, therefore b=3)
a(a+3)(a-3)=80
There are a few ways to do this, but 3 factors of 80 that are fairly close will all be around cuberoot(80), between 4 and 5. Factors near there include 1, 2, 4, 5, 8, 10. Clearly, 2, 5, and 8 fit.