Puzzle for August 29, 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
eq.6 may be written as: F = (A + B + C + D + E) ÷ 5 Multiply both sides of the above equation by 5: 5 × F = 5 × (A + B + C + D + E) ÷ 5 which becomes eq.6a) 5×F = A + B + C + D + E
Hint #2
In eq.1, replace A + B + C + D + E wih 5×F (from eq.6a): 5×F + F = 30 which makes 6×F = 30 Divide both sides of the above equation by 6: 6×F ÷ 6 = 30 ÷ 6 which makes F = 5
Hint #3
In eq.2, substitute 5 for F: eq.2a) B = 5 - A
Hint #4
In eq.5, substitute 5 for F: eq.5a) E = A + 5
Hint #5
Substitute (5 - A) for B (from eq.2a) in eq.4: D = A - (5 - A) which becomes D = A - 5 + A which makes eq.4a) D = 2×A - 5
Hint #6
Substitute A + 5 for E (from eq.5a), and (5 - A) for B (from eq.2a) in eq.3: C = A + 5 - (5 - A) which becomes C = A + 5 - 5 + A which makes C = 2×A
Hint #7
Substitute 5 - A for B (from eq.2a), 2×A for C, 2×A - 5 for D (from eq.4a), A + 5 for E (from eq.5a), and 5 for F in eq.1: A + 5 - A + 2×A + 2×A - 5 + A + 5 + 5 = 30 which simplifies to 5×A + 10 = 30 Subtract 10 from each side of the equation above: 5×A + 10 - 10 = 30 - 10 which makes 5×A = 20 Divide both sides by 5: 5×A ÷ 5 = 20 ÷ 5 which makes A = 4
Solution
Since A = 4, then: B = 5 - A = 5 - 4 = 1 (from eq.2a) C = 2×A = 2 × 4 = 8 D = 2×A - 5 = 2×4 - 5 = 8 - 5 = 3 (from eq.4a) E = A + 5 = 4 + 5 = 9 (from eq.5a) and ABCDEF = 418395