What is the smallest positive integer $n$ for which $9n-2$ and $7n + 3$ share a common factor greater than $1$?

  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.)

Check

  9(23) -2 = 205 = 5×41

  7(23) +3 = 164 = 4×41

square pyramid

step-by-step explanation:

square prism looks like nothing like that

triangular prism has a rectangular base instead of a square base

tetrahedron has a triangular base