Question #9c261

1 Answer
sente
May 5, 2016

#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#

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