Puzzle for May 12, 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
Add B to both sides of eq.3: B + E + B = A – B + B which becomes eq.3a) 2×B + E = A In eq.4, replace A with 2×B + E (from eq.3a): D – (2×B + E – E) = 2×B + E – E which becomes D – 2×B = 2×B Add 2×B to both sides of the above equation: D – 2×B + 2×B = 2×B + 2×B which makes D = 4×B
Hint #2
Subtract E from each side of eq.3a: 2×B + E – E = A – E which becomes 2×B = A – E In eq.5, replace A – E with 2×B: eq.5a) C ÷ D = 2×B
Hint #3
In eq.5a, substitute 4×B for D: C ÷ 4×B = 2×B Multiply both sides of the above equation by 4×B: C ÷ 4×B × 4×B = 2×B × 4×B which makes C = 8×B²
Hint #4
Substitute 4×B for D in eq.6: A ÷ (4×B – B) = (4×B – B) which becomes A ÷ 3×B = 3×B Multiply both sides of the equation above by 3×B: A ÷ 3×B × 3×B = 3×B × 3×B which makes A = 9×B²
Hint #5
Substitute 9×B² for A, and 4×B for D in eq.2: 9×B² – 4×B = B + 4×B which becomes 9×B² – 4×B = 5×B Add 4×B to both sides of the above equation: 9×B² – 4×B + 4×B = 5×B + 4×B which makes 9×B² = 9×B Divide both sides by 9×B (B ≠ 0, since D ≠ 0 (from eq.5), and 4×B = D): 9×B² ÷ 9×B = 9×B ÷ 9×B which makes B = 1 making A = 9×B² = 9 × 1² = 9 × 1 = 9 C = 8×B² = 8 × 1² = 8 × 1 = 8 D = 4×B = 4 × 1 = 4
Hint #6
Substitute 1 for B, and 9 for A in eq.3a: 2×1 + E = 9 which becomes 2 + E = 9 Subtract 2 from each side of the above equation: 2 + E – 2 = 9 – 2 which makes E = 7
Solution
Substitute 8 for C, 1 for B, and 7 for E in eq.1: 8 + F = 1 + 7 which becomes 8 + F = 8 Subtract 8 from each side of the equation above: 8 + F – 8 = 8 – 8 which makes F = 0 and makes ABCDEF = 918470