This is from a youtube video on the Chinese Remainder Theorem - https://www.youtube.com/watch?v=ru7mWZJlRQg
What the author has done thus far is basically
1.Make sure that the mods, 3, 4, 5 are relatively pairwise prime by showing that gcd(3,4) = 1, gcd(3,5) = 1 and gcd(4,5) = 1.
2.Set up a table with mod 3, mod 4, mod 5 as the columns. He multiplied one column by the other two so that when applied it's modulus, say for mod 3, mod 4 and mod 5 column values will be set to zero.
3.Here's the part that I have a question about. The author states that the first linear congruence
x $\equiv$ 2(mod 3) must be satisfied and to do so, he mods all the values by 3. The only non zero value will be the value in column 1(because of the last step).
My question is by the definition of a is congruent to b modulo m(below)
Shouldn't the author have to subtract all the values by 2 first and then mod 3, that way he get 18 mod 3, 13 mod 3, and 9 mod 3, or 0, 1, and 0? Is there a reason he doesn't have to do this? To me, this isn't consistent with the definition of congruency