Question

H.C.F. of 85 and 119 can be expressed in the form of $85x-153$. Value of $x$ is :

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

This problem combines finding the H.C.F. of two numbers with solving a linear equation. The Euclidean Algorithm is an efficient method for finding the H.C.F. of larger numbers.

Answer 1

The Correct Option is B

Step 1: Find the H.C.F. (Highest Common Factor) of 85 and 119 using the Euclidean Algorithm or prime factorization.
Method 1: Euclidean Algorithm
Divide 119 by 85: $119 = 85 \times 1 + 34$ Divide 85 by the remainder 34: $85 = 34 \times 2 + 17$ Divide 34 by the remainder 17: $34 = 17 \times 2 + 0$ The last non-zero remainder is 17. So, H.C.F. (85, 119) = 17. Method 2: Prime Factorization
Prime factors of 85: $5 \times 17$
Prime factors of 119: $7 \times 17$
The common factor is 17.
So, H.C.F. (85, 119) = 17.
Step 2: Set up the given equation.
We are given that the H.C.F. can be expressed in the form $85x - 153$.
From Step 1, H.C.F. = 17.
So, we have the equation:
$85x - 153 = 17$
Step 3: Solve the equation for $x$.
Add 153 to both sides of the equation:
$85x = 17 + 153$
$85x = 170$
Divide both sides by 85:
$x = \frac{170}{85}$
$x = 2$
Step 4: Verify the answer (optional).
Substitute $x=2$ back into the expression $85x - 153$:
$85(2) - 153 = 170 - 153 = 17$.
This matches the H.C.F., so the value of $x$ is correct.
Step 5: Compare with the given options.
The calculated value of $x$ is 2, which matches option (2). (2) 2