If the cube root of 1331 is 11 and the cube root of 1728 is 12, what is the cube root of 1442897?

(Do this one in your head - no writing or calculators allowed!)

The number is greater than 1.331*10^6 but less than 1.728*10^6 and so; the cube root must lie between 111 and 119 inclusively.....(i)

Since the given positive integer is congruent to 7 when reduced in modulo 10, it follows that the required cube root must possess the form 3 ( Mod 10).

Consequently, from (i) it now follows that the required cube root must be 113.

*Edited on ***September 12, 2007, 11:28 am**