Question

suppose m,n are the integers and m= n²-n. Then show that m²-2m is divisible by 24​

Answer 1

• m^2 = n^4 + n^2 - 2n^3
• 2m = 2n^2 - 2n
• m^2 - 2m = n^4 - n^2 - 2n^3 + 2n
• = n(n^3 - n - 2n^2 + 2)
• n = 4 x + 0,1,2, or 3
• check for each

Answer 2

• m=n²-n

⇒m²-2m=(n²-n)²-2(n²-n)

              =n⁴+n²-2n³-2n²+2n

              =n⁴-2n³-n²+2n

              =n³(n-2)-n(n-2)

              =(n³-n)(n-2)

              =n(n²-1)(n-2)

              =n(n-1)(n+1)(n-2)

              =(n-2)(n-1)n(n+1)

• The above expression represents the product of 4 consecutive numbers,∴it should be divisible by 24 provided n>2.