Question

If two positive integers a and b are written as $a={{x}^{3}}{{y}^{2}}$ and $b=x{{y}^{3}}$ ; x, y are prime numbers, then HCF(a,b) is
(a). $xy$
(b). $x{{y}^{2}}$
(c). ${{x}^{3}}{{y}^{3}}$
(d). ${{x}^{2}}{{y}^{2}}$

Answer 1

Hint: To determine the HCF of the numbers, express the number in terms of the product of its prime factors and find the product of their common factors. This is the method of prime factorization.

Complete step by step answer:
Before proceeding with the solution, let’s understand the concept of prime factorization. A prime number is a number which is not divisible by any other number except 1 and itself. Any number can be expressed as a product of prime numbers. All the prime numbers, which when multiplied, give a product equal to a number (say x) are called the prime factors of the number x. To write the prime factors of a number, we should always start with the smallest prime number, i.e. 2 and check divisibility. If the number is divisible by the prime number, then we write the number as a product of the prime number and another number, which will be the quotient when the given number is divided by the prime number. Then, we take the quotient and repeat the same process. This process is repeated till we are left with 1 as the quotient.
For example: Consider the number 51. It is an even number. So, it is not divisible by 2. The sum of the digits of 51 is 5 + 1 = 6. Hence, 51 is divisible by 3. Now, $51=3\times 17$ . Now, we take 17. We know, 17 is a prime number. Hence, the prime factors of 51 are 3 and 17.
Now a can also be written as $x\times x\times x\times y\times y$ in terms of its prime factors. We can also express b in terms of its prime factors as $x\times y\times y\times y$ . Now as we are asked to find the HCF, we will find the product of the common prime factors of a and b.
$\therefore HCF=x\times y\times y=x{{y}^{2}}$ .
Therefore, the value of HCF(a, b) is $x{{y}^{2}}$ . Hence, the answer to the above question is option (b).


Note: The above method can only be used when it is mentioned that x and y are prime, as if it is not mentioned, x and y may have some common factors, which would affect the HCF of the numbers.