Question

Question #9c261

Answer 1

`x=28`

Explanation:

`{(21-=385" (mod x)"), (587 -= 167" (mod x)"):}`

`<=>{(364-=0" (mod x)"), (420 -= 0" (mod x)"):}`

(by subtracting `21` from each side of the first congruency, and `167` from each side of the second)

`<=> {(364 = k_1x" for some "k_1inZZ), (420 = k_2x" for some "k_2inZZ):}`

Thus, as all of the above steps are reversible, `x` is the greatest integer which divides both `364` and `420`. We can find this using the Euclidean algorithm:

`420 = 1xx364 + 56`

`364 = 6xx56 + 28`

`56 = 2xxcolor(red)(28) + 0`

Thus `x = "gcd"(420,364) = 28`