Question

if m is an odd integer then show that 24/m( m2-1)​

Answer 1

Step-by-step explanation:

if m is an odd no. then it can be written a.

let another variable b=4

by division algorithm,

a=bq + r , where r=remainder and q=quotient

Now,

a=4q + r, where r can be =0,1,2,3

therfore a=4q + 1[ since a is odd]

=4q + 3

therefore a^2=m^2

(a^2) -1 =(4q + 1)^2 -1=(16q^2 + 1 + 8q)-1= 16q^2 + 8q

=8(2q^2 + q) -(Eq.1)

(a^2) -1= (4q +3)^2

=(16q^2 +9 + 24q) -1

=16q^2 +8 + 24q

=8(2q^2 +3q +1) -(Eq.2)

..since eq.1 and eq.2 are divisible by 8 ...

therefore m^2 -1 will be divisible by 8