Question

If n and t are positive integers, what is the greatest prime factor of the product nt ? (1) The greatest common factor of n and t is 5. (2) The least common multiple of n and t is 105.

Answer 1

Greatest prime factor of
`nt` is 7.

Explanation:

Given that

Two positive integers are
`n` and
`t`.

(1) Greatest Common Factor or HCF of
`n` and
`t` is 5.

(2) Least Common Multiple or LCM of
`n` and
`t` is 105.

To find:

The greatest prime factor of the product
`nt` = ?

Solution:

First of all, let us learn about a property of HCF and LCM of two numbers.

The product of two numbers
`p` and
`q` is equal to the product of their HCF and LCM.

`p * q =LCM* HCF`

Using this property for the given numbers:

`n* t =5* 105\\OR\\nt =5*105`

Now, let us make prime factors of
`5 * 105` to find the greatest of the prime factors.

`5* 105 = 5* 5 * 21 =5* 5 * 3 * \bold{7}`

So, the prime factors of
`5 * 105` are 5, 5, 3 and 7.

Greatest prime factor of
`nt` is 7.