Puzzle for May 27, 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
Subtract D from each side of eq.5: B – D – D = A + D – D which becomes B – 2×D = A In eq.4, replace A with B – 2×D: B – 2×D = D + F Add 2×D to both sides of the equation above: B – 2×D + 2×D = D + F + 2×D which becomes eq.4a) B = 3×D + F
Hint #2
In eq.3, replace B with 3×D + F (from eq.4a): C + E + F = 3×D + F Subtract F from each side of the equation above: C + E + F – F = 3×D + F – F which becomes eq.3a) C + E = 3×D
Hint #3
In eq.3a, substitute C + D for E (from eq.2): C + C + D = 3×D which becomes 2×C + D = 3×D Subtract D from both sides of the equation above: 2×C + D – D = 3×D – D which becomes 2×C = 2×D Divide both sides by 2: 2×C ÷ 2 = 2×D ÷ 2 which makes C = D
Hint #4
Substitute C for D in eq.2: E = C + C which makes E = 2×C
Hint #5
Substitute C for D, and 2×C for E in eq.6: C + 2×C = A – 2×C Add 2×C to both sides of the above equation: C + 2×C + 2×C = A – 2×C + 2×C which makes 5×C = A
Hint #6
Substitute 5×C for A, and C for D in eq.4: 5×C = C + F Subtract C from each side of the equation above: 5×C – C = C + F – C which makes 4×C = F
Hint #7
Substitute 2×C for E, and 4×C for F in eq.3: C + 2×C + 4×C = B which makes 7×C = B
Solution
Substitute 5×C for A, 7×C for B, C for D, 2×C for E, and 4×C for F eq.1: 5×C + 7×C + C + C + 2×C + 4×C = 20 which simplifies to 20×C = 20 Divide both sides by 20: 20×C ÷ 20 = 20 ÷ 20 which makes C = 1 making A = 5×C = 5 × 1 = 5 B = 7×C = 7 × 1 = 7 D = C = 1 E = 2×C = 2 × 1 = 2 F = 4×C = 4 × 1 = 4 and ABCDEF = 571124