Question

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

Answer 1

Hint: This number is based on the concept of prime numbers and the greatest common divisor. So what we will do is we will find the factors of 30 p and 14 p and find which common factor is the greatest common factor.

Complete step-by-step solution:
The greatest common factor is the highest factor that divides two or more numbers. For example, let us take two numbers, say 9 and 15. Factors of any number are those numbers which on dividing the given numbers give positive integer value. So, The factors of 9 are 1, 3, 9. And the factors of 15 are 1, 3, 5, 15.
Now, the greatest common factor will be the largest factor which is common in both number’s prime factorization. And, if we carefully see, that number is 3. So, G.C.D of 9 and 15 is equal to 3.
Now, on factorization of 30 p, the factors will be equal to 2, 3, 5, p.
\[\begin{align}
  & 2\left| \!{\underline {\,
  \text{ }30 \,}} \right. p \\
 & 3\left| \!{\underline {\,
  \text{ 15p} \,}} \right. \\
 & 5\left| \!{\underline {\,
  \text{ 5p} \,}} \right. \\
 & p\left| \!{\underline {\,
  \text{ p} \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  \text{ 1} \,}} \right. \\
\end{align}\]
And, on factorization of 14 p, the factors will be equal to p, 2, 7, 14.
\[\begin{align}
  & 2\left| \!{\underline {\,
  \text{ 14} \,}} \right. p \\
 & 7\left| \!{\underline {\,
  \text{ 7p} \,}} \right. \\
 & p\left| \!{\underline {\,
  \text{ p} \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  \text{ 1} \,}} \right. \\
\end{align}\]
Here, factors are multiplied with p is because as mentioned in the question, p is a prime and prime number can never be factored, as prime numbers are divisible by 1 and itself.
As $p > 0$, the only greatest common divisor we have between 30 p and 14 p is 2 p.
Hence, we can say that the only greatest common divisor we have between 30 p and 14 p is 2 p.

Note: For, doing these types of question one must have the knowledge of the greatest common factors which that largest factor common for both numbers, what are prime numbers which are those numbers, which are divisible by 1 and itself, and how to find factors of any number by dividing the number by its prime factors unless we get 1 as remainder. The only factors of ‘ a ‘, if a is a prime number is 1 and ‘ a ’ only.