Question

Find the HCF of 130 and 91 and express it in the form 130x + 91y, where x and y are integers. Also prove that this expression is not unique

Answer 1

Given,

130 and 91 are two numbers.

To find,

HCF of 130 and 91 and express it in the form of 130x + 91y, where x and y are integers.

To prove,

130x + 91y is not unique.

prime factors of 130 = 2 × 5 × 13

prime factors of 91 = 7 × 13

common factors of 130 and 91 = 13

so, HCF of 130 and 91 = 13

now 130x + 91y = 13

⇒10x + 7y = 1

if we take x = 5 and y = -7

then, 10 × 5 + 7 × -7 = 50 - 49 = 1 [satisfied]

so, x = 5 and y = -7 is a solution.

again, we take x = -2, y = 3

then, 10 × -2 + 7 × 3 = -20 + 21 = 1 [satisfied]

so, x = -2 and y = 3 is another solution.

it can be expressed as 130 × 5 + 91 × -7 or 130 × -2 + 91 × 3.

here is clear that given expression is not unique (there are two solutions).