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I need a source for questions like these:

  • What is the remainder when $3^{5555}$ is divided by 80?

  • Find the remainder when $25^{100} + 11^{500}$ is divided by 3.

with worked solutions.

I have searched online, and I keep finding single examples, but I need something like between 10 and 20 exercises, and without worked solutions I have no way of knowing how I'm doing or what I'm getting stuck on. Any suggestions please?

Robin Andrews
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  • Do you understand that for the first two you need Euler's Theorem (https://en.wikipedia.org/wiki/Euler%27s_theorem) while for the latter two you need Bezout's Identity (https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity)? – Andrea Mori Nov 12 '22 at 11:30
  • Nope. I thought they were more simple than that. I'm looking for questions using basic rules of congruence arithmetic. I've edited the question. – Robin Andrews Nov 12 '22 at 11:31
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    Since $25\equiv 1 \pmod 3$, $25^{100}\equiv 1 \pmod 3$. The other part of the second problem is similarly straight forward. For the first one, use the Chinese Remainder theorem. Otherwise, you'll find many problems of this form on this site, and I'd also look up the Art of Problem Solving series. – lulu Nov 12 '22 at 11:38
  • @RobinAndrews: to solve the problem "find the remainder of $a^N$ divided by $m$" you have to find a value $k$ such that the remainder of $a^k$ divided by $m$ is $1$. This allows to simplify greatly the exponent $N$ and perform easily the computation. For small values of $m$ this can be easily done by hand (see lulu's comment). For bigger values of $m$ Euler's Theorem provides an answer when $a$ and $m$ have no common divisors. – Andrea Mori Nov 12 '22 at 11:47
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    for the first problem use the fact that $3^4=81=1$ mod $80$ – Lozenges Nov 12 '22 at 12:01
  • The number in the second exercise is the sum of two squares not divisible by $3$. Such squares are congruent to $1$ mod $3$ hence the sum is congruent $2$ mod $3$. – Peter Nov 12 '22 at 14:09
  • Use the linked mod order reduction with $!\bmod 80!:\ 3^4\equiv 1,,$ & $!\bmod 3!:\ 25\equiv 1,\ 11\equiv -1,\ (-1)^2\equiv 1\ \ $ – Bill Dubuque Nov 12 '22 at 15:13
  • I'm surprised that no has attempted to answer the actual question I asked, and the question has been closed on the basis of it being a different question from what it actually is. My question was not ambiguous IMO, and was about where to find practice exercises, not how to solve them. I appreciate the attempt to help. – Robin Andrews Nov 12 '22 at 16:11
  • Why not just make them up. If you know how to solve them it should be easy to do. – Roddy MacPhee Nov 13 '22 at 21:01
  • Because I wouldn't be sure of the answers. My understanding is "in progress." – Robin Andrews Nov 14 '22 at 16:13

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