Puzzle for February 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
Add D to both sides of eq.6: D + F + D = B – D + D which becomes 2×D + F = B In eq.5, replace B with 2×D + F: F – D = 2×D + F – F which becomes F – D = 2×D Add D to both sides of the equation above: F – D + D = 2×D + D which makes F = 3×D
Hint #2
In eq.6, replace F with 3×D: D + 3×D = B – D which becomes 4×D = B – D Add D to both sides of the above equation: 4×D + D = B – D + D which makes 5×D = B
Hint #3
In eq.3, substitute 5×D for B, and 3×D for F: C = 5×D + 3×D which makes C = 8×D
Hint #4
In eq.4, substitute 5×D for B, and 3×D for F: A = 5×D + D + 3×D which makes A = 9×D
Hint #5
Substitute 8×D for C in eq.2: E = 8×D + D which makes E = 9×D
Solution
Substitute 9×D for A and E, 5×D for B, 8×D for C, and 3×D for F in eq.1: 9×D + 5×D + 8×D + D + 9×D + 3×D = 35 which simplifies to 35×D = 35 Divide both sides of the above equation by 35: 35×D ÷ 35 = 35 ÷ 35 which means D = 1 making A = E = 9×D = 9 × 1 = 9 B = 5×D = 5 × 1 = 5 C = 8×D = 8 × 1 = 8 F = 3×D = 3 × 1 = 3 and ABCDEF = 958193