# If a number ‘x’ is divisible by another number ‘y’, then ‘x’ is also divisible by all the prime factors of ‘y’.

By BYJU'S Exam Prep

Updated on: September 25th, 2023

The axiom “if a number ‘x’ is divisible by another number ‘y’, then ‘x’ is also divisible by all the prime factors of ‘y'” is absolutely true. A natural number other than 1 with only the number 1 and itself as factors is known as a prime factor. To understand this, let us take x to be divisible by y, whose quotient is z.

So, x/y = z

or x = y × z (equation 1)

‘y’ can be expressed as a product of its primes as follows:

y = p × q × r, where the prime factors of y are p × q × r (equation 2)

Putting the values of equation 2 in equation 1, we will get

x = p × q × r × z (equation 3)

x/p = z × q × r

x/r = z × p × q

Therefore, all the prime factors of ‘y’ can divide ‘x’.

## If ‘x’ is divisible by ‘y’ then ‘x’ is divisible by all prime factors of ‘y’

The process of expressing a number as the product of its prime factors is known as prime factorization. The prime factorization of 24 is 2 × 2 × 2 × 3 wherein 2 and 3 are the prime factors of 24. Similarly the prime factorization of 980 is 2 × 2 × 5 × 7 × 7. So, the prime factors of 980 are 2, 5, and 7. Below are some key points related to the numbers:

• If a number can be divided by another number, it can also be divided by each of the other number’s factors.
• Co-prime numbers are two numbers that only share the number 1 as a factor.
• The smallest prime number that is even is 2. Except for 2, all prime numbers are odd.
• A prime number is any number that has only two factors, namely 1, and the number itself. Composite numbers are those that have more than two factors. Neither prime nor composite, number one is not.

Summary:

## If a number ‘x’ is divisible by another number ‘y’, then ‘x’ is also divisible by all the prime factors of ‘y’.

The statement “if a number ‘x’ is divisible by another number ‘y’, then ‘x’ is also divisible by all the prime factors of ‘y’” is completely true. Prime factorization is the technique of representing a number as the product of its prime factors.

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