Question

Show that 5n cannot end with the digit 2

Answer 1

General proof

IF THE NUMBER 5n, FOR ANY n, WERE TO BE DIVISIBLE BY 2,THEN IT WOULD HAVE 2 IN ITS PRIME FACTORISATION.

THAT IS, THE PRIME FACTORISATION OF 5nWOULD CONTAIN THE PRIME NO 2.THIS IS, NOT POSSIBLE BECAUSE FACTORS OF 5n IS (5)n

SO, THE PRIME FACTORISATION OF 5nCONTAINS ONLY AND ONLY 5 NOT 2. SO THE UNIQUENESS OF THE FUNDAMENTAL THEOREM OF ARITHMETIC GURANTEES THAT THERE ARE NO OTHER PRIMES IN THE FACTORISATION OF 5n.

SO, THERE IS NO NATURAL NUMBER n FOR WHICH 5n IS DIVISIBLE BY 2

With an example
5n
i) if n=1 then 5n =5*1=5
ii) if n=2 then 5n=5*2=10
iii)if n=3 then 5n=5*3=15
iv) if n=4 then 5n=5*4=20
Like for all. Natural values of n, 5n ends with 0 or 5
It never ends with 2