Puzzle for January 3, 2022 ( )
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.6, replace A with B + D (from eq.3): B + E = B + D – E In the above equation, subtract B from both sides, and add E to both sides: B + E – B + E = B + D – E – B + E which simplifies to 2×E = D
Hint #2
In eq.4, replace D with 2×E: C = 2×E + E which makes C = 3×E
Hint #3
In eq.5, substitute 3×E for C: F = 3×E + E which makes F = 4×E
Hint #4
Substitute 3×E for C, and 4×E for F in eq.2: B = 3×E + 4×E which makes B = 7×E
Hint #5
Substitute 7×E for B, and 2×E for D in eq.3: A = 7×E + 2×E which makes A = 9×E
Solution
Substitute 9×E for A, 7×E for B, 3×E for C, 2×E for D, and 4×E for F in eq.1: 9×E + 7×E + 3×E + 2×E + E + 4×E = 26 which simplifies to 26×E = 26 Divide both sides of the above equation by 26: 26×E ÷ 26 = 26 ÷ 26 which means E = 1 making A = 9×E = 9 × 1 = 9 B = 7×E = 7 × 1 = 7 C = 3×E = 3 × 1 = 3 D = 2×E = 2 × 1 = 2 F = 4×E = 4 × 1 = 4 and ABCDEF = 973214