Question

What is the common factor of \[5xy\] and\[15{y^2}\]?
A.$5y$
B.$x$
C.$3$
D.${y^2}$

Answer 1

Hint: A prime number is a number which is greater than one and is not a product of two smaller numbers (both greater than one) or a number which has only two factors one and the number itself termed as prime number. HCF full form is the highest common factor which means the greatest number which divides given two or more numbers is known as H.C.F of them. Use these definitions to solve the problem.
In case of algebraic expressions, say $6{x^3}y{z^2}$ and\[3xyz\], we first find the HCF of the constant parts (3 and 6). This would clearly be 3. Now, we find the terms of both algebraic expressions. Clearly,\[xyz\] is common to both the terms. Thus, HCF in this case becomes \[3xyz\]. Thus, by observing the stated example we can say that the highest common factor (HCF) is the expression which is a factor of both the algebraic expression.

Complete step-by-step answer:
We will find HCF for both numbers by writing the prime factorization of each of the numbers. We will choose the common prime factors of all the two algebraic expressions. The number and variable obtained by choosing the common prime factors will be the HCF of two algebraic expressions.
The prime factorization of \[5xy\] is = $5 \times x \times y$.
The prime factorization of \[15{y^2}\] is = $5 \times 3 \times {y^2}$.
The terms which are common to prime factorization of both the numbers is $5 \times y = 5y$.
Thus, the HCF of \[5xy\] and \[15{y^2}\] is \[5y\].
So, the common factor of \[5xy\] and \[15{y^2}\] is \[5y\].
Therefore, option (A) is the correct answer.

Note: We have to study two important aspects when dealing with these kinds of questions:
i.The product of the two algebraic expressions is equal to the product of their prime factors.
ii.The product of the two algebraic expressions is equal to the product of their L.C.M. and H.C.F.