What is the smallest positive integer $n$ for which $9n-2$ and $7n + 3$ share a common factor greater than $1$?
1 answer:
Answer:
23
Step-by-step explanation:
You can use Euler's method to find the GCF of these values:
(9n -2) mod (7n +3) = 2n -5
(7n +3) mod (2n -5) = n +18
(2n -5) mod (n +18) = n -23
We want this to be zero, so n = 23. (Note that none of the other remainders are zero for any positive integer n.)
The smallest positive integer n for which (9n-2) and (7n+3) share a common factor is 23. (Their common factor is 41.)
_____
<em>Check</em>
9(23) -2 = 205 = 5×41
7(23) +3 = 164 = 4×41
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