Modular Arithmetic Exercises
Published:
Find the least positive value of $x$ such that
(i) $71 \equiv x \pmod{8}$
(ii) $78 + x \equiv 3 \pmod{5}$
(iv) $96 \equiv \frac{x}{7} \pmod{5}$
(v) $5x \equiv 4 \pmod{6}$
If $x$ is congruent to 13 modulo 17 then $7x - 3$ is congruent to...
Solve $5x \equiv 4 \pmod{6}$
Solve $3x - 2 \equiv 0 \pmod{11}$
What is the time 100 hours after 7 a.m.?
What is the time 15 hours before 11 p.m.?
Today is Tuesday. My uncle will come after 45 days. In which day is my uncle coming?
Prove that $2^n + 6 \times 9^n$ is always divisible by 7 for any positive integer $n$.
Find the remainder when $2^{81}$ is divided by 17.
Animated Video Solution
The first half plays free, the full solution is in the app.
Step by Step Written Solution
Hi Pushpa, let's solve this interesting modular arithmetic problem together. We're looking for the remainder when two to the power of eighty-one is divided by seventeen.
Problem Summary
Find the remainder: $2^{81} \pmod{17}$
To solve this efficiently, we'll use properties of congruences and power cycles. Let's start by finding a power of two that is close to a multiple of seventeen.
Step 1: Find a base power
Notice that sixteen is extremely useful because it is congruent to negative one modulo seventeen. This will simplify our calculations significantly.
So we can write sixteen as negative one modulo seventeen.
Now, we need to express our original exponent, eighty-one, in terms of our base exponent, four. Using the division algorithm, we know that eighty-one equals four times twenty plus one.
Step 2: Rewrite the exponent
The rest of this solution is on Solvi
5 more steps are locked. Watch the full animated, narrated solution for free.
Snap a photo, solve any question like this.
Watch the Rest for Free
Free to download · First solutions are on us
