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
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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
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Verified answer
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).