Puzzle for April 8, 2020 ( )
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.5, replace F with A + E (from eq.2): C + E + A + E = A + D which becomes C + 2×E + A = A + D Subtract A from each side of the equation above: C + 2×E + A – A = A + D – A which becomes eq.5a) C + 2×E = D
Hint #2
In eq.5a, replace D with B + C + E (from eq.3): C + 2×E = B + C + E Subtract C and E from each side of the above equation: C + 2×E – C – E = B + C + E – C – E which simplifies to E = B
Hint #3
In eq.4, replace D with B + C + E (from eq.3): B + B + C + E + E = A + C which becomes 2×B + C + 2×E = A + C Subtract C from both sides of the above equation: 2×B + C + 2×E – C = A + C – C which becomes eq.4a) 2×B + 2×E = A
Hint #4
In eq.4a, substitute B for E: 2×B + 2×B = A which makes 4×B = A
Hint #5
Substitute 4×B for A, and B for E in eq.2: F = 4×B + B which makes F = 5×B
Hint #6
Substitute 4×B for A, B for E, and 5×B for F in eq.6: 4×B – B = D – 5×B which becomes 3×B = D – 5×B Add 5×B to both sides of the equation above: 3×B + 5×B = D – 5×B + 5×B which makes 8×B = D
Hint #7
Substitute 8×B for D, and B for E in eq.3: 8×B = B + C + B which becomes 8×B = 2×B + C Subtract 2×B from each side fo the equation above: 8×B – 2×B = 2×B + C – 2×B which makes 6×B = C
Solution
Substitute 4×B for A, 6×B for C, 8×B for D, B for E, and 5×B for F in eq.1: 4×B + B + 6×B + 8×B + B + 5×B = 25 which simplifies to 25×B = 25 Divide both sides of the equation above by 25: 25×B ÷ 25 = 25 ÷ 25 which means B = 1 making A = 4×B = 4 × 1 = 4 C = 6×B = 6 × 1 = 6 D = 8×B = 8 × 1 = 8 E = B = 1 F = 5×B = 5 × 1 = 5 and ABCDEF = 416815