Question

Find the prime factorisation of the following numbers:
${85}^3-{68}^3+5^3-{22}^3$

Answer 1

Let us consider $68-5+22=85$, therefore, the given problem can be rewritten as:

$\left[\right(68)+(-5)+(22){]}^3-(68{)}^3-(-5{)}^3-(22{)}^3$

We know the identity: $(a+b+c{)}^3-a^3-b^3-c^3=3(a+b\left)\right(b+c\left)\right(c+a)$

Using the above identity taking $a=68$, $b=-5$ and $c=22$, the equation

$\left[\right(68)+(-5)+(22){]}^3-(68{)}^3-(-5{)}^3-(22{)}^3$ can be factorised as follows:

$\left[\right(68)+(-5)+(22){]}^3-(68{)}^3-(-5{)}^3-(22{)}^3=3(68-5)(-5+22)(22+68)=3\times 63\times 17\times 90$

$=2\times (3\times 3\times 3\times 3\times 3)\times 5\times 7\times 17=2\times 3^5\times 5\times 7\times 17$

Hence, ${85}^3-{68}^3+5^3-{22}^3=2\times 3^5\times 5\times 7\times 17$