hello, could you help with this prove:

given that p is a prime and p|aⁿ, prove that pⁿ|aⁿ


thanks, janna

The "Fundamental Theorem of Arithmetic" (a rather grand name, so you know it is important) says that every positive integer greater than one can be represented uniquely as a product of one or more prime numbers, ignoring order of the factors.

That means that if you have the number 1710, it can be represented as 2 x 3 x 3 x 5 x 19, all of those factors are prime numbers. No other prime numbers go into 1710. Since 3 shows up twice, a more common way to show this is: 2 x 3² x 5 x 19

Now let's think about 1710 as represented by "a" in your expression. That means that aⁿ is the same as the product of all the same primes above - each taken to the nth power:
1710ⁿ = 2ⁿ x 3²ⁿ x 5ⁿ x 19ⁿ.

Your question says that p (a prime) divides aⁿ. Remember that no other primes besides the ones we've listed go into aⁿ (it's unique). Can you see why that means that p also divides a itself? And since it is prime, then it must be equal to 2, 3, 5, or 19 (if we use the example of a=1710).

You can also see pretty clearly that if taking a number to the nth power is the same as taking each of its prime factors to the nth power, then pⁿ is one of those new factors.

Hope this helps!
Stephen La Rocque>