Puzzle for August 16, 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.4, replace E with B + C (from eq.2): B + B + C = A + C which becomes 2×B + C = A + C Subtract C from both sides of the above equation: 2×B + C – C = A + C – C which makes 2×B = A
Hint #2
In eq.6, replace A with 2×B: E – B = 2×B + B which becomes E – B = 3×B Add B to both sides of the above equation: E – B + B = 3×B + B which makes E = 4×B
Hint #3
In eq.2, substitute 4×B for E: 4×B = B + C Subtract B from each side of the above equation: 4×B – B = B + C – B which makes 3×B = C
Hint #4
Substitute 3×B for C, and 4×B for E in eq.3: 3×B + 4×B = B + D which becomes 7×B = B + D Subtract B from each side of the equation above: 7×B – B = B + D – B which makes 6×B = D
Hint #5
Substitute 2×B for A, and 6×B for D in eq.5: 2×B + 6×B = B + F which becomes 8×B = B + F Subtract B from each side of the equation above: 8×B – B = B + F – B which makes 7×B = F
Solution
Substitute 2×B for A, 3×B for C, 6×B for D, 4×B for E, and 7×B for F in eq.1: 2×B + B + 3×B + 6×B + 4×B + 7×B = 23 which simplifies to 23×B = 23 Divide both sides of the above equation by 23: 23×B ÷ 23 = 23 ÷ 23 which means B = 1 making A = 2×B = 2 × 1 = 2 C = 3×B = 3 × 1 = 3 D = 6×B = 6 × 1 = 6 E = 4×B = 4 × 1 = 4 F = 7×B = 7 × 1 = 7 and ABCDEF = 213647