Thread: Finding a unique cube root

Hello, pranay!

If we have an integer (say ) which ends in either 1, 3, 7 or 9,
then how can one find another integer having at most the same number of digits as
but when cubed would end in exactly the same digit as .

E.g. .if a = 123, then the required number is 927
because 123 ends in 3 and 923^3 also ends in 3.
. . It could be any 3-digit number (or smaller) ending in 7.

Another example: .a = 435621, then required number is 786941.
. . It could be any 6-digit number (or smaller) ending in 1.

Just make a list of the endings of cubes . . .

. . a ends in: . . a^3 ends in:
. . - - - - - - - - - - - - - - - - - -
. . . . . 0. . . . . . . . . 0
. . . . . 1. . . . . . . . . 1
. . . . . 2. . . . . . . . . 8
. . . . . 3. . . . . . . . . 7
. . . . . 4. . . . . . . . . 4
. . . . . 5. . . . . . . . . 5
. . . . . 6. . . . . . . . . 6
. . . . . 7. . . . . . . . . 3
. . . . . 8. . . . . . . . . 2
. . . . . 9. . . . . . . . . 9



Example: .a = _ 4

. . .Then: .b = _ 4


Example: .a = _ _ 3

. . .Then: .b = _ _ 7

Thanks for that, but what if we need to find the smallest cube root which when cubed would end in the given number?
like for the first example 927^3 would end in 123?