Question

can you help me doing this guys.h ere the question in the above pic

Answer 1

 Let a be any positive integer. Then the prime factorization of a is as follows : a = p1 p2 … pn , where p1 , p2 , …., pn are primes, not necessarily distinct. Therefore a 2 = (p1 p2 … pn ) (p1 p2 … pn ) = p 2 1 p 2 2 … p2 n . Now, here we have been given that p divides a 2 . Therefore, from the Fundamental Theorem of Arithmetic, it follows that p is one of the prime factors of a 2 . Also, using the uniqueness part of the Fundamental Theorem of Arithmetic, we realise that the only prime factors of a 2 are p1 p2 … pn . So p is one of p1, p2, … pn . Now, since p is one of p1 p2 … pn , p therefore, it divides a.

Now,

p=2,p=5

a=p1p2=5x2=10
a²=10x10=100
Now, here we have been given that p divides a 2 . Therefore, from the Fundamental Theorem of Arithmetic, it follows that p is one of the prime factors of a 2 . Also, using the uniqueness part of the Fundamental Theorem of Arithmetic, we realise that the only prime factors of a 2 are p1 p2 … pn . So p is one of p1, p2, … pn . Now, since p is one of p1 p2 … pn , p therefore, it divides a.