Question

Find the value of each letter in the sums shown. Each letter stands for a digit between 0 and 9 (inclusive).

Show me the correct steps.
X + X + X = BX
XXX + YYY + ZZZ = ABCD​

Answer 1

• The sums are: 15 and 2109

-----------------------

First, we exclude zero for X, Y, Z, A and B, because the first digit can't be zero.

From the first equation we get:

• X + X + X = BX
• 3X = 10B + X (B is the number of tens)
• 2X = 10B
• X = 5B

The only possible solution is:

• X = 5 and B = 1

From the second equation we get:

• XXX + YYY + ZZZ = ABCD​
• X*11 + Y*111 + Z*111 = ABCD
• 111(X + Y + Z) = ABCD

Let's call X + Y + Z = k, then:

• 111k is the 4-digit number with all different digits.

We can try all values of k, between:

• k = 1 + 2 + 3 = 6 and
• k = 9 + 8 + 7 = 24

The only 111k product with 4-different digits, or with the second digit of 1 (B = 1 from the first equation) is:

• 111*19 = 2109

We already have X = 5, so:

• Y + Z = 19 - 5
• Y + Z = 14

Possible values of Y and Z are 6 and 8 (or 8 and 6), not possible pairs are 9 and 5, also 7 and 7 due to repeats.

Therefore, the solution for all letters is:

• X = 5, Y = 6, Z = 8, A = 2, B = 1, C = 0, D = 9 or
• X = 5, Y = 8, Z = 6, A = 2, B = 1, C = 0, D = 9