The L.C.M. of two numbers is 495 and their H.C.F. is 5. If the sum of the numbers is 100, then their difference is equal to:
A) 90
B) 46
C) 10
D) 70

Question

High School

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Answer :

Parrain Parrain · Answer 1 · 2024-05-12T04:09:09-04:00

The difference of the numbers would be equal to C. 10.

The product of two numbers is equal to the product of their LCM and HCF. So, we have:

[tex]\[a \times b = \text{LCM} \times \text{HCF}\]\[a \times b = 495 \times 5\]\[a \times b = 2475\][/tex]

We also know that the sum of the two numbers is 100. So, we have:

a + b = 100

System of equations:

1. a x b = 2475

2. a + b = 100

Solve this system of equations to find the values of a and b.

We know that:

[tex]\[ (a + b)^2 = a^2 + b^2 + 2ab \][/tex]

Substituting the given values:

[tex]\[ (100)^2 = a^2 + b^2 + 2(2475) \]\[ 10000 = a^2 + b^2 + 4950 \]\[ a^2 + b^2 = 10000 - 4950 \]\[ a^2 + b^2 = 5050 \][/tex]

Now, we know that:

[tex]\[ (a - b)^2 = a^2 + b^2 - 2ab \][/tex]

Substituting the known values:

[tex]\[ (a - b)^2 = 5050 - 2(2475) \]\[ (a - b)^2 = 5050 - 4950 \]\[ (a - b)^2 = 100 \][/tex]

Taking the square root of both sides:

[tex]\[ a - b = \sqrt{100} \]\[ a - b = 10 \][/tex]

The difference between the two numbers is 10.