If d = 0, 1, 4, 5, 6, or 9, then d^3 ends in d.
If d = 2, 3, 7, or 8, then d^3 ends in 10-d.
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How to mentally compute the cube root of the cube of a two digit number.
Thread (1/10)
If d = 0, 1, 4, 5, 6, or 9, then d^3 ends in d.
If d = 2, 3, 7, or 8, then d^3 ends in 10-d.
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For example, if 50653 is the cube of an integer, it's the cube of a number ending in 7.
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Then 10a <= n <= 10(a+1), and so n^3 is between (10a)^3 and (10(a+1))^3, i.e. between 1000 a^3 and 1000 (a+1)^3.
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Chop off the last three digits of x.
Then a, the first digit of n, is the largest number whose cube is no greater than what's left of x.
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Lopping off the last three digits gives 50.
3^3 < 50 < 4^3, so the first digit of the cube root of 50653 is 3.
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