Theorem: Let p be a prime number. If p divides
, then p divides a, where a is a positive integer.
Solution:
Let
where, p1, p2, p3, …, pn are prime numbers which are necessarily not distinct.
It is given that p divides
. From the Fundamental theorem of Arithmetic, we know that every composite number can be expressed as product of unique prime numbers. This means that p is one of the numbers from (p1.p2.p3.p4.p5……pn).
We have a=(p1.p2.p3.p4.p5..pn) and p is one of the numbers from (p1.p2.p3.p4.p5……pn).
It means that p also divides a.
Hence, it is proved that if p divides
then it also divides a.
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