how many natural numbers m are such that: 2202-=2022(modm)

Answer 1

The statement

2202 ≡ 2022 (mod m) (1)

means that there exist natural numbers a, b, r, so that

2202 = ma + r (2)

2022 = mb + r (3)

0 ≤ r < m (4)

From (2) and (3), a necessary condition is that there exit a, b so that

2202 - ma = 2022 - mb

Simplify into

180 = m(a-b)

That means a necessary condition is that m divides 180.

The answer to the original given question are all the divisors of 180.

You should be suspicious of the last statement because being a divisor of 180

is only a necessary condition, not a sufficient one.

For that we also need (4).

You could assure yourself that all the divisors of 180 actually satisfy (1)

by checking them all explicitly.

But you do not need to do that for the following reason.

Suppose that m is a divisor of 180.

That means there exists natural number q so that

180 = mq

Let

a = floor(2202/m)

b = a - q

r = 2202 - ma

By algebra you can see that the numbers a, b, r have been defined so that all

(2), (3), (4) are satisfied.