Question

The LCM and HCF of two positive numbers are 175 and 5 respectively. If the sum of the numbers is 60. What is the difference between them ?

Answer 1

Hint: Since the HCF is 5 we can take the numbers to be 5a and 5b. And using the rule Product of two numbers = Product of their HCF and LCM, we get ab = 35, hence we discuss the possibilities and we get two options and now we need to check whether it satisfies the condition that the sum of the numbers is 60. On checking we get the correct pair and further we need to find the difference between them.

Step by step solution :
We are given that the LCM and HCF of two positive numbers are 175 and 5
Since we are given that the highest common factor of the two numbers is 5
Let's assume the numbers to be 5a and 5b
Now we know that
Product of two numbers = Product of their HCF and LCM
$
   \Rightarrow 5a\times 5b = 175\times 5 \\
   \Rightarrow 25ab = 175\times 5 \\
   \Rightarrow ab = \dfrac{{175\times 5}}{{25}} = 35 \\
 $
Now we get the product of two numbers a and b to be 35
Let's discuss the various possibilities
The possibilities of getting the product 35 are $1\times 35{\text{ and }}5\times 7$
Now we get the two set of values for 5a and 5b to be
$ \Rightarrow \left( {5,175} \right){\text{ and }}\left( {25,35} \right)$
Now we are given that the sum of the numbers is 60
And we can see that only the pair $\left( {25,35} \right)$ satisfies that condition
Hence now the difference between the numbers is given by
$ \Rightarrow 35 - 25 = 10$

Therefore the difference between the numbers is 10.

Note :
1) The H.C.F of two or more numbers is smaller than or equal to the smallest number of given numbers.
2) The L.C.M of two or more numbers is greater than or equal to the greatest number of given numbers.
3) The smallest number which is exactly divisible by x, y and z is L.C.M of x, y, z.