Quiz Linear Combinations

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Quiz 1: GCDs as Linear Combinations
1. Warm-Up. Fill in the boxes. Feel free to do some scratch work off to the side.
• Let’s find x, y ∈ Z such that gcd(324, 392) = 324x + 392y.
• First, use the Euclidean Algorithm to find gcd(324, 392).
392
68
52
16
=
=
=
=
=
&middot; 324 + 68
4 &middot; + 52
1 &middot; 52 + 3&middot;+4
4 &middot; + 0.
Thus, gcd(324, 392) = .
• Next, rewrite the information so that each remainder is isolated:
16
52
68
=
=
=
=
52 − 3 &middot; 16
− 1 &middot; 52
324 − &middot; 68
− 1 &middot; 324
• Now, get (which is the gcd(324, 392)) to become expressed in terms of 324 and
392.
4 =
=
=
=
=
=
=
52 − &middot; 16
52 − 3 &middot; ( − 1 &middot; 52)
4 &middot; 52 − 3 &middot; 68
4 &middot; (324 − 4 &middot; 68) − &middot; 68
4 &middot; − 19 &middot; 68
4 &middot; 324 − 19 &middot; (392 − 1 &middot; 324)
23 &middot; − 19 &middot; 392.
• Finally, this tells us that = gcd(324, 392) = 324x + 392y where x = and y = .
TURN OVER →
2. Using the Euclidean Algorithm, find the greatest common divisor of the given integers,
and then write it as a linear combination of those integers.
That is, given a, b ∈ Z, find gcd(a, b) and then find integer values of x and y so that
gcd(a, b) = ax + by.
(a) a = 496, b = 192. CHECK YOUR WORK CAREFULLY :o
```