Puzzle for October 4, 2021 ( )
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 C + F (from eq.4): C + F – E = C + E In the equation above, subtract C from both sides, and add E to both sides: C + F – E – C + E = C + E – C + E which makes F = 2×E
Hint #2
In eq.5, replace F with 2×E: E + 2×E = D – 2×E which becomes 3×E = D – 2×E Add 2×E to both sides of the above equation: 3×E + 2×E = D – 2×E + 2×E which makes 5×E = D
Hint #3
In eq.3, substitute 5×E for D: 5×E + E = B which makes 6×E = B
Hint #4
Substitute 6×E for B in eq.2: 6×E + E = C which makes 7×E = C
Hint #5
Substitute 7×E for C, and 2×E for F in eq.4: 7×E + 2×E = A which makes 9×E = A
Solution
Substitute 9×E for A, 6×E for B, 7×E for C, 5×E for D, and 2×E for F in eq.1: 9×E + 6×E + 7×E + 5×E + E + 2×E = 30 which simplifies to 30×E = 30 Divide both sides of the above equation by 30: 30×E ÷ 30 = 30 ÷ 30 which means E = 1 making A = 9×E = 9 × 1 = 9 B = 6×E = 6 × 1 = 6 C = 7×E = 7 × 1 = 7 D = 5×E = 5 × 1 = 5 F = 2×E = 2 × 1 = 2 and ABCDEF = 967512