Puzzle for April 20, 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.5, replace E with B + F (from eq.3), and D with A + C (from eq.2): A + B + F = C + A + C + F which becomes A + B + F = A + 2×C + F Subtract A and F from each side of the above equation: A + B + F – A – F = A + 2×C + F – A – F which simplifies to B = 2×C
Hint #2
eq.4 may be written as: D = C + B + F In the equation above, replace B + F with E (from eq.3): eq.4a) D = C + E
Hint #3
In eq.2, substitute C + E for D (from eq.4a): C + E = A + C Subtract C from both sides of the equation above: C + E – C = A + C – C which makes E = A
Hint #4
Substitute A for E, and 2×C for B into eq.6: A + F – 2×C = A + 2×C + C which becomes A + F – 2×C = A + 3×C In the above equation, subtract A from both sides, and add 2×C to both sides: A + F – 2×C – A + 2C = A + 3C – A + 2C which makes F = 5×C
Hint #5
Substitute 2×C for B, and 5×C for F in eq.3: E = 2×C + 5×C which makes E = 7×C and also makes A = E = 7×C
Hint #6
Substitute 7×C for A in eq.2: D = 7×C + C which makess D = 8×C
Solution
Substitute 7×C for A and E, 2×C for B, 8×C for D, and 5×C for F in eq.1: 7×C + 2×C + C + 8×C + 7×C + 5×C = 30 which simplifies to 30×C = 30 Divide both sides of the above equation by 30: 30×C ÷ 30 = 30 ÷ 30 which means C = 1 making A = E = 7×C = 7 × 1 = 7 B = 2×C = 2 × 1 = 2 D = 8×C = 8 × 1 = 8 F = 5×C = 5 × 1 = 5 and ABCDEF = 721875