How does this "Guess the number" trick work?

Question

Algorithms

I have a book by Gershenzon " Professor Jigsaw Puzzles ".
Among tricks like "Think of a number", there is one.

I tried to deduce the equation to understand how it works, but it seems that 7th grade knowledge is not enough for this (or I'm dumb).
Can anyone explain?

Answer 1

We thought of a number: N
Remainders from division: A, B, C
Factors a, b, c - integers

N = A + 3a
N = B + 4b
N = C + 5c

A = N - 3a
B = N - 4b
C = N - 5c

40A + 45B + 36C = 
40(N - 3a) + 45(N - 4b) + 36(N - 5c) = 
40N - 120a + 45N - 180b + 36N - 180c = 
121N - 120a - 180b - 180c = 
N + 120N - 120a - 180b - 180c

120N, 120a, 180b, 180c obviously divisible by 60 evenly Remainder
after dividing by 60: N

Answer 2

In my opinion, here the whole explanation is exhaustively stated after the words

The rule is very simple