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

Answer:

Greatest prime factor of $nt$ is 7.

Step-by-step 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.

$p \times q =LCM\times HCF$

Using this property for the given numbers:

$n\times t =5\times 105\\OR\\nt =5\times105$

Now, let us make prime factors of $5 \times 105$ to find the greatest of the prime factors.

$5\times 105 = 5\times 5 \times 21 =5\times 5 \times 3 \times \bold{7}$

So, the prime factors of $5 \times 105$ are 5, 5, 3 and 7.

Greatest prime factor of $nt$ is 7.