Puzzle for June 7, 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
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Hint #1
In eq.5, replace F with D + E (from eq.2): D + D + E = C + E which becomes 2×D + E = C + E Subtract E from both sides of the above equation: 2×D + E – E = C + E – E which makes 2×D = C
Hint #2
In eq.4, replace C with 2×D: 2×D + D = B – D which becomes 3×D = B – D Add D to both sides of the equation above: 3×D + D = B – D + D which makes 4×D = B
Hint #3
In eq.6, substitute 4×D for B: 4×D – D = A – 4×D which becomes 3×D = A – 4×D Add 4×D to both sides of the above equation: 3×D + 4×D = A – 4×D + 4×D which makes 7×D = A
Hint #4
Substitute 7×D for A in eq.3: E = 7×D + D which makes E = 8×D
Hint #5
Substitute 8×D for E in eq.2: F = D + 8×D which makes F = 9×D
Solution
Substitute 7×D for A, 4×D for B, 2×D for C, 8×D for E, and 9×D for F in eq.1: 7×D + 4×D + 2×D + D + 8×D + 9×D = 31 which simplifies to 31×D = 31 Divide both sides of the above equation by 31: 31×D ÷ 31 = 31 ÷ 31 which means D = 1 making A = 7×D = 7 × 1 = 7 B = 4×D = 4 × 1 = 4 C = 2×D = 2 × 1 = 2 E = 8×D = 8 × 1 = 8 F = 9×D = 9 × 1 = 9 and ABCDEF = 742189