Puzzle for March 21, 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
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Hint #1
In eq.6, replace F with A + E (from eq.2): A + E – A = A – E which becomes E = A – E Add E to both sides of the above equation: E + E = A – E + E which makes 2×E = A
Hint #2
In eq.2, replace A with 2×E: F = 2×E + E which makes F = 3×E
Hint #3
In eq.5, substitute 2×E for A, and 3×E for F: D = 2×E + E + 3×E which makes D = 6×E
Hint #4
Substitute 6×E for D in eq.3: B = 6×E + E which makes B = 7×E
Hint #5
Substitute 7×E for B in eq.4: C = 7×E + E which makes C = 8×E
Solution
Substitute 2×E for A, 7×E for B, 8×E for C, 6×E for D, and 3×E for F in eq.1: 2×E + 7×E + 8×E + 6×E + E + 3×E = 27 which simplifies to 27×E = 27 Divide both sides of the above equation by 27: 27×E ÷ 27 = 27 ÷ 27 which means E = 1 making A = 2×E = 2 × 1 = 2 B = 7×E = 7 × 1 = 7 C = 8×E = 8 × 1 = 8 D = 6×E = 6 × 1 = 6 F = 3×E = 3 × 1 = 3 and ABCDEF = 278613