Number Theory

Chinese remainder

Combine congruences, or see exactly why they clash.

For checking your work after you’ve tried it yourself.

Step 1:
Add the conditions

Each one is x = r (mod m). Up to 6.

Step 2:
Press Combine

Or press Enter.

Step 3:
Read the answer

All solutions at once, or exactly where they clash.
What it does.
Combines conditions like "x leaves remainder 2 when divided by 3" into a single condition that describes every x satisfying them all. If they contradict each other, it shows where.
When it helps.
Checking the answer to a system of remainders you solved by hand.
What it won’t do.
Up to 6 conditions, with moduli up to 101210^{12}.
Example.
x≡2(mod3)x \equiv 2 \pmod 3, x≡3(mod5)x \equiv 3 \pmod 5 and x≡2(mod7)x \equiv 2 \pmod 7 give x≡23(mod105)x \equiv 23 \pmod{105}.
Congruences x≡r(modm)x \equiv r \pmod{m}
x≡x \equiv(mod)
x≡x \equiv(mod)

2 to 6 rows. m from 1 to 101210^{12}; r can be any whole number.

Try