If $y = 30 p$ , and $p$ is prime, what is the greatest common factor of $y$ and $14 p$ , in terms of $p$ ?

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Solution

The greatest common factor of y (= 30 p) and 14p is the product of all the common prime factors, using the lower power of repeated factors.
The only repeated factors are 2 and p: $2^{1} \times p^{1} = 2 \times p = 2 p$ . Again, you would get the same answer if p were any positive integer.
Number 2 3 5 7 p
30p $2^{1}$ $\times$ $3^{1}$ $\times$ $5^{1}$ - $\times$ $p^{1}$
14p $2^{1}$ - - $\times$ $7^{1}$ $\times$
$p^{1}$
GCE $2^{1}$ $p^{1}$

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Number	2		3		5		7		p
30p	$2^{1}$	$\times$	$3^{1}$	$\times$	$5^{1}$		-	$\times$	$p^{1}$
14p	$2^{1}$		-		-	$\times$	$7^{1}$	$\times$	$p^{1}$
GCE	$2^{1}$								$p^{1}$

If y=30p, and p is prime, what is the greatest common factor of y and 14p, in terms of p?

If $y = 30 p$ , and $p$ is prime, what is the greatest common factor of $y$ and $14 p$ , in terms of $p$ ?