# Finding a unique cube root

• Apr 19th 2011, 07:13 AM
pranay
Finding a unique cube root
If we have an integer(say $a$) 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 $a$ but when cubed would end in exactly the same digit as $a$.
E.g if a = 123
then the required number is 927 because 123 ends in 3 and 927^3 also ends in 3 .
Other examples:
a = 435621 then required number is 786941
Thanks.
• Apr 19th 2011, 09:12 AM
Soroban
Hello, pranay!

Quote:

If we have an integer (say $\,a$) 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 $\,a$
but when cubed would end in exactly the same digit as $\,a$.

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

• Apr 19th 2011, 09:34 AM
pranay
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?