Puzzle for January 28, 2019 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
Scratchpad
Help Area
Hint #1
In eq.5, replace C - D with F (from eq.4): A - B = F - F which means A - B = 0 Add B to both sides of the equation above: A - B + B = 0 + B which means A = B
Hint #2
In eq.3, replace B with A: A + D = C Substitute A + D for C in eq.4: A + D - D = F which makes A = F
Hint #3
In eq.2, replace both B and F with A: A + A = D - A Add A to each side: A + A + A = D - A + A which means 3×A = D
Hint #4
Substitute A for both B and F, and 3×A for D in eq.5: A - A = C - 3×A - A which simplifies to 0 = C - 4×A Add 4×A to both sides: 0 + 4×A = C - 4×A + 4×A which makes 4×A = C
Hint #5
Substitute A for F, 4×A for C, and 3×A for D in eq.6: E - A = A + 4×A + 3×A which becomes E - A = 8×A Add A to both sides: E - A + A = 8×A + A which makes E = 9×A
Solution
Substitute A for B and F, 4×A for C, 3×A for D, and 9×A for E in eq.1: A + A + 4×A + 3×A + 9×A + A = 19 which simplifies to 19×A = 19 Divide both sides by 19: 19×A ÷ 19 = 19 ÷ 19 which means A = 1 making B = F = A = 1 C = 4×A = 4 × 1 = 4 D = 3×A = 3 × 1 = 3 E = 9×A = 9 × 1 = 9 and ABCDEF = 114391