# Solving the congruence relation $90\equiv 6 \pmod n$

A family member of mine is currently taking high school Algebra 2 and is learning about the basics of modular arithmetic. The following question was asked of the students:

Find all integers n that satisfy the following:

$$90\equiv 6 \pmod n$$

The family member asked me for help, as she was stuck. But I too was stumped. It has been two years since I took Algebra 2, and I barely remember modular arithmetic. I seem to remember that $$90\equiv 6 \pmod n$$ means that 90 mod n = 6 mod n -- however, I am not sure how that can help me solve the problem.

I know this is a rather elementary question so I apologize if it has already been answered. That said, I have not been able to find an answer so far, as most sources deal with equations where the unknown value is something other than the one "inside" the mod parenthesis.

• $90\equiv6\pmod n$ means $n$ divides $90-6$ Commented Sep 23, 2019 at 3:34
• The definition is surely given in their textbook. What does it say? Commented Sep 23, 2019 at 3:40
• @JyrkiLahtonen I don't think they have a textbook, alas...
– Will
Commented Sep 23, 2019 at 3:41

$$90\equiv6\pmod n$$ means $$n$$ divides $$90-6,$$ so the solutions for $$n$$ are the factors of $$84.$$

• Thanks! Why does it mean n divides 90 - 6, though?
– Will
Commented Sep 23, 2019 at 3:38
• That's the definition, so $90=qn+r$ and $6=pn+r$ where $0\le r<n$ Commented Sep 23, 2019 at 3:41