[kazinator]

Modulo 9, a decimal integer is equal to the sum of its digits. More precisely, congruent, notated by ≡:

For instance 123 ≡ 1+2+3 ≡ 6 (mod 9).

We show congruences using ≡, and always have (mod N) on the far right to indicate the modulus for the congurence.

More generally, if ABC is a decimal string, then we know that ABC ≡ A + B + C (mod 9).

Moreover ABC + DEF + GHI must be congruent to A+B+C + D+E+F + G+H+I (mod 9).

And if ABC + DEF + GHI is equal to 123J, then A+B+C + D+E+F + G+H+I ≡ 123J ≡ 1+2+3+J (mod 9).

Thus:

A+B+C+D+E+F+G+H+I ≡ 1+2+3+J (mod 9)

Now suppose we add J to both sides:

A+B+C+D+E+F+G+H+I+J ≡ 1+2+3+J+J (mod 9)

OK so now we know that the left hand side A+...+J contains all elements from 0 to 9, because of the problem constraint that the letters represent unique digits. The numbers 0 to 9 add together to 45. Now 45 is congruent to 0 (mod 9).

Therefore:

A+B+C+D+E+F+G+H+I+J ≡ 45 ≡ 0 ≡ 6 + 2J (mod 9)

We no longer care about the A+..+J; it's vanished. We solve the remaining equation:

0 ≡ 6 + 2J (mod 9)

If 6 + X (mod 9) is congruent to 0, X must be one of {... -6, 3, 12, 21, 30, 39 ...}: the set of integers congruent to 3, (mod 9).

If X = 2J, where J is a one-digit decimal integer, X must be an even, non-negative integer. That rules out -6, 3 and 21. It can't be 30, because J can't be 15. X must be 12, which gives J = 6.

[quietbritishjim]

That was all good, except this bit was not obvious to me:

> Modulo 9, a decimal integer is equal to the sum of its digits. More precisely, congruent, notated by ≡:

> For instance 123 ≡ 1+2+3 ≡ 6 (mod 9).

This is the part where the parent comment's explanation was helpful.

[kazinator]

There is a nice, succinct explanation for this (which relies on some intuition or knowledge of modulo arithmetic and congruences).

First, a small defnition: let dsum(X) denote "sum of the digits of the decimal representation of whole number x".

In the modulo 9 congruence, 0 and 9 are equivalent symbols. 9 ≡ 0 (mod 9). Though distinct integers outside of the congruence they are indistinguishable elements of the congruence.

Observe that since 9 is congruent to 0, the number 10 is congruent to 1: 10 ≡ 1 (mod 9). Therefore, all powers of 10 are also congruent to 1: 1 ≡ 10 ≡ 100 ≡ 1000 ... (mod 9).

Next, note that these numbers x ∈ {1, 10, 100, 1000, ...} all have the property that x ≡ dsum(X) (mod 9). The sum of decimal digits of each power of 10 is 1, and each power of 10 is congruent to 1, modulo 9.

Within a congruence, whenever we multiply by any congruence element that is indistinguishable from 1, we get the same element. So all of these numbers 1, 10, 100, ... are identity elements in the modulo 9 congruence.

Identity means that, for instance 7 ≡ 70 ≡ 700 ≡ 7000 = 7000 ... (mod 9). If we multiply the congruence element 7 by any power of 10, the result is congruent to 7, modulo 9, because 10 is the identity element: no matter how many times we multiply an integer by 10, the resulting integer is the same modulo 9 congruence element. (Moreover, also note that dsum(7) = dsum(70) = dsum(700) ...).

Next, note that every decimal integer is formed by a sum combination of multiples of power of 10.

For instance, 1234 is equal to 1 x 1000 + 2 x 100 + 3 x 10 + 4.

But suppose we do this arithmetic in the modulo 9 congruence: it must necessarily be that:

1000 + 200 + 30 + 4 ≡ 1 + 2 + 3 + 4 (mod 9).

This is because, individually, 1000 ≡ 1 (mod 9), and 200 ≡ 2 (mod 9), 30 = 3 (mod 9). The power of 10 factors do not matter, since they are really powers of the identity element 1 in the mod 9 congruence.

The succinct explanation is then: every whole number has a decimal form made up of digits, which are multiplied by powers of 10. But in the modulo 9 congruence, powers of 10 do not matter: they map to the multiplicative identity element 1. That leaves the sum of the digits indistinguishable (in the modulo 9 congruence) from the value of the number (which is also a kind of sum of the digits, with each digit being scaled by a different power of 10).