Question

show that n^2(n-1)^2 is divisible by 4 when n is any positive number.​

Answer 1

Step-by-step explanation:

let n be even no 2a, a>0

then n^2=(2a)^2=4a^2

(n-1)^2=(2a -1)^2

n^2×(n-1)^2=4a^2×(2a-1)^2

=4×positive no.×positive no.

as square of any number is positive no.

product is clearly multiple of 4

let n be odd no, 2a+1, a>0

n^2= (2a+1)^2

n-1 =2a+1-1=2a

(n-1)^2=(2a)^2=4a^2

n^2×(n-1)^2=(2a+1)^2×4×a^2

=4×positive no.× positive no

product is multiple of 4

so product is multiple of 4, when n is even or odd

hence it is divisible by 4

hope this helps