Question

Given that a is a multiple of 456, find the greatest common divisor of 3a^3+a^2+4a+57 and a.

Answer 1

Answer: 57

Explanation:

Given: a is a multiple of 456

Let
`a = 456 x`

Then, expression
`3a^3+a^2+4a+57 =3(456x)^3+(456x)^2+4(456x)+57`

Since 456 = 57 x 8

Then,
`3(456x)^3+(456x)^2+4(456x)+57=3(57* 8x)^3+(57* 8x)^2+4(57* 8x)+57`

`=3(57)^3* (8x)^3+(57)^2* (8x)^2+4(57)* (8x)+57`

Taking 57 out as common

`=57[3(57)^2* (8x)^3+(57)* (8x)^2+4* (8x)+1]`

Now, the greatest common divisor of
`a = 456 x` and
`3a^3+a^2+4a+57=57[3(57)^2* (8x)^3+(57)* (8x)^2+4* (8x)+1]` is 57.

Hence, the greatest common divisor of 3a^3+a^2+4a+57 and a is 57.