Q37 of 40 Page 1

Show that (12)n cannot end with digit 0 or 5 for any natural number n.

OR


Prove that (√2 + √5) is irrational.


If the number 12n, for any natural number n, ends with the digit 0 or 5, then it is divisible by 5.

That is, the prime factorization of 12n contains the prime 5.


This is not possible because prime factorisation of


12n = (22 × 3)n = 22n × 3n


So, the only primes in the factorisation of 12n are 2 and 3


And according to the fundamental theorem of arithmetic guarantees that there are no other primes in the factorization of 12n. So, there is no natural number n for which 12n ends with the digit zero.


OR


Let √2+√5 be a rational number.


A rational number can be written in the form of p/q where p,q are integers.


√2+√5 = p/q


Squaring on both sides,


(√2+√5)2 = (p/q)2


√22+√52+2(√5)(√2) = p2/q2


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