Puzzle for December 5, 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.2, replace C with B + E (from eq.3): B + B + E = A + E which becomes 2×B + E = A + E Subtract E from each side of the equation above: 2×B + E - E = A + E - E which makes 2×B = A
Hint #2
In eq.6, replace A with 2×B: F = 2×B + B which makes F = 3×B
Hint #3
In eq.4, substitute 2×B for A, and 3×B for F: D = 2×B + 3×B which makes D = 5×B
Hint #4
Substitute 5×B for D, and 3×B for F in eq.5: E = 5×B + 3×B which makes E = 8×B
Hint #5
Substitute 8×B for E in eq.3: C = B + 8×B which makes C = 9×B
Solution
Substitute 2×B for A, 9×B for C, 5×B for D, 8×B for E, and 3×B for F in eq.1: 2×B + B + 9×B + 5×B + 8×B + 3×B = 28 which simplifies to 28×B = 28 Divide both sides of the above equation by 28: 28×B ÷ 28 = 28 ÷ 28 which means B = 1 making A = 2×B = 2 × 1 = 2 C = 9×B = 9 × 1 = 9 D = 5×B = 5 × 1 = 5 E = 8×B = 8 × 1 = 8 F = 3×B = 3 × 1 = 3 and ABCDEF = 219583