Answer:
Greatest prime factor of [tex]nt[/tex] is 7.
Step-by-step explanation:
Given that
Two positive integers are [tex]n[/tex] and [tex]t[/tex].
(1) Greatest Common Factor or HCF of [tex]n[/tex] and [tex]t[/tex] is 5.
(2) Least Common Multiple or LCM of [tex]n[/tex] and [tex]t[/tex] is 105.
To find:
The greatest prime factor of the product [tex]nt[/tex] = ?
Solution:
First of all, let us learn about a property of HCF and LCM of two numbers.
The product of two numbers [tex]p[/tex] and [tex]q[/tex] is equal to the product of their HCF and LCM.
[tex]p \times q =LCM\times HCF[/tex]
Using this property for the given numbers:
[tex]n\times t =5\times 105\\OR\\nt =5\times105[/tex]
Now, let us make prime factors of [tex]5 \times 105[/tex] to find the greatest of the prime factors.
[tex]5\times 105 = 5\times 5 \times 21 =5\times 5 \times 3 \times \bold{7}[/tex]
So, the prime factors of [tex]5 \times 105[/tex] are 5, 5, 3 and 7.
Greatest prime factor of [tex]nt[/tex] is 7.
