Puzzle for April 10, 2023 ( )
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.4, replace E with A + D (from eq.2): A + D - F = D + F In the above equation, subtract D from both sides, and add F to both sides: A + D - F - D + F = D + F - D + F which makes eq.4a) A = 2×F
Hint #2
In eq.3, replace A with 2×F: F - C = 2×F - F which becomes F - C = F Subtract F from each side of the equation above: F - C - F = F - F which makes -C = 0 which means C = 0
Hint #3
In eq.5, substitute 2×F for A, and 0 for C: D + F = 2×F + 0 - D which becomes D + F = 2×F - D In the above equation, subtract F from both sides, and add D to both sides: D + F - F + D = 2×F - D - F + D which becomes 2×D = F
Hint #4
Substitute (2×D) for F in eq.4a: A = 2×(2×D) which makes A = 4×D
Hint #5
Substitute 4×D for A in eq.2: E = 4×D + D which makes E = 5×D
Hint #6
Substitute 5×D for E, and 0 for C in eq.6: B - 5×D = 0 - D + 5×D which becomes B - 5×D = 4×D Add 5×D to both sides of the above equation: B - 5×D + 5×D = 4×D + 5×D which makes B = 9×D
Solution
Substitute 4×D for A, 9×D for B, 0 for C, 5×D for E, and 2×D for F in eq.1: 4×D + 9×D + 0 + D + 5×D + 2×D = 21 which simplifies to 21×D = 21 Divide both sides of the above equation by 21: 21×D ÷ 21 = 21 ÷ 21 which means D = 1 making A = 4×D = 4 × 1 = 4 B = 9×D = 9 × 1 = 9 E = 5×D = 5 × 1 = 5 F = 2×D = 2 × 1 = 2 and ABCDEF = 490152