Puzzle for August 21, 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.6, replace C with B + F (from eq.5): B + B + F = E + F which becomes 2×B + F = E + F Subtract F from each side of the equation above: 2×B + F - F = E + F - F which makes 2×B = E
Hint #2
In eq.2, replace E with 2×B: F = B + 2×B which makes F = 3×B
Hint #3
In eq.5, substitute 3×B for F: C = B + 3×B which makes C = 4×B
Hint #4
Substitute 4×B for C, and 2×B for E in eq.4: A = 4×B + 2×B which makes A = 6×B
Hint #5
Substitute 6×B for A, and 3×B for F in eq.3: D = 6×B + 3×B which makes D = 9×B
Solution
Substitute 6×B for A, 4×B for C, 9×B for D, 2×B for E, and 3×B for F in eq.1: 6×B + B + 4×B + 9×B + 2×B + 3×B = 25 which simplifies to 25×B = 25 Divide both sides of the above equation by 25: 25×B ÷ 25 = 25 ÷ 25 which means B = 1 making A = 6×B = 6 × 1 = 6 C = 4×B = 4 × 1 = 4 D = 9×B = 9 × 1 = 9 E = 2×B = 2 × 1 = 2 F = 3×B = 3 × 1 = 3 and ABCDEF = 614923