Puzzle for October 11, 2021 ( )
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.3, replace F with B + C (from eq.2): C + E = B + B + C which becomes C + E = 2×B + C Subtract C from each side of the above equation: C + E – C = 2×B + C – C which makes E = 2×B
Hint #2
In eq.5, replace E with 2×B: D – 2×B = B + 2×B which becomes D – 2×B = 3×B Add 2×B to both sides of the equation above: D – 2×B + 2×B = 3×B + 2×B which makes D = 5×B
Hint #3
In eq.4, substitute 5×B for D, and 2×B for E: A = B + 5×B + 2×B which makes A = 8×B
Hint #4
In eq.6, substitute 5×B for D, and 2×B for E: F = 5×B + 2×B which makes F = 7×B
Hint #5
Substitute 7×B for F in eq.2: B + C = 7×B Subtract B from each side of the equation above: B + C – B = 7×B – B which makes C = 6×B
Solution
Substitute 8×B for A, 6×B for C, 5×B for D, 2×B for E, and 7×B for F in eq.1: 8×B + B + 6×B + 5×B + 2×B + 7×B = 29 which simplifies to 29×B = 29 Divide both sides of the above equation by 29: 29×B ÷ 29 = 29 ÷ 29 which means B = 1 making A = 8×B = 8 × 1 = 8 C = 6×B = 6 × 1 = 6 D = 5×B = 5 × 1 = 5 E = 2×B = 2 × 1 = 2 F = 7×B = 7 × 1 = 7 and ABCDEF = 816527