Question4. Assuming that , prove the following: (b) or 3 .
Studdy Solution
STEP 1
1. , meaning and are coprime.
2. We need to prove that is either or .
STEP 2
1. Use properties of gcd and linear combinations.
2. Prove that any common divisor must divide a specific integer.
3. Conclude based on the divisibility condition.
STEP 3
We start by considering a common divisor of and . By definition, divides both expressions.
STEP 4
Express the gcd condition using linear combinations. Since divides both and , it must also divide any linear combination of these two expressions.
Consider the expression:
Thus, divides .
STEP 5
Since divides both and (because it divides ), and we know , must divide any integer combination of and .
STEP 6
Now, consider the expression:
Thus, divides . Since , must divide .
STEP 7
Since divides , the possible values for are or .
Therefore, we have shown that or .
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