Puzzle for August 1, 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.
* "F mod E" equals the remainder of (F ÷ E).
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Hint #1
Add C to both sides of eq.5: F ÷ E – C + C = B + C – D + C which becomes F ÷ E = B + 2×C – D Since B, C, and D are all integers, then in the equation above: B + 2×C – D = an integer, which means F ÷ E = an integer
Hint #2
Since F ÷ E is an integer, then: F = an integer multiple of E, which means remainder (F ÷ E) = 0 which means F mod E = 0 In eq.6, replace F mod E with 0: C = 0
Hint #3
eq.4 may be written as: D = (B + C + F) ÷ 3 Multiply both sides of the above equation by 3: D × 3 = ((B + C + F) ÷ 3) × 3 which becomes D × 3 = B + C + F In the above equation, replace C with 0: D × 3 = B + 0 + F which is the same as eq.4a) 3×D = B + F
Hint #4
In eq.2, substitute 0 for C: 0 + E + F = B + D which becomes E + F = B + D Subtract E from each side of the above equation: E + F – E = B + D – E which becomes eq.2a) F = B + D – E
Hint #5
Substitute B + D – E for F (from eq.2a) in eq.4a: 3×D = B + B + D – E which becomes 3×D = 2×B + D – E Subtract D from both sides of the above equation: 3×D – D = 2×B + D – E – D which becomes eq.4b) 2×D = 2×B – E
Hint #6
Add D to both sides of eq.3: B – D + E + D = D + F – B – C + D which becomes B + E = 2×D – B – C + F Substitute 2×B – E for 2×D (from eq.4b), and 0 for C into the above equation: B + E = 2×B – E – B – 0 + F which becomes B + E = B – E + F Subtract B from both sides, and add E to both sides: B + E – B + E = B – E + F – B + E which makes 2×E = F
Hint #7
Substitute 2×E for F, and 0 for C in eq.5: 2×E ÷ E – 0 = B + 0 – D which becomes 2 = B – D Add D to both sides of the above equation: 2 + D = B – D + D which becomes eq.5a) 2 + D = B
Hint #8
Substitute (2 + D) for B (from eq.5a) into eq.4b: 2×D = 2×(2 + D) – E which becomes 2×D = 4 + 2×D – E In the above equation, subtract 2×D from both sides, and add E to both sides: 2×D – 2×D + E = 4 + 2×D – E – 2×D + E which simplifies to E = 4 making F = 2×E = 2 × 4 = 8
Hint #9
Substitute 8 for F, 2 + D for B (from eq.5a), and 4 for E in eq.2a: 8 = 2 + D + D – 4 which becomes 8 = 2×D – 2 Add 2 to both sides of the equation above: 8 + 2 = 2×D – 2 + 2 which makes 10 = 2×D Divide both sides by 2: 10 ÷ 2 = 2×D ÷ 2 which makes 5 = D making B = 2 + D = 2 + 5 = 7 (from eq.5a)
Solution
Substitute 7 for B, 0 for C, 5 for D, 4 for E, and 8 for F in eq.1: A + 7 + 0 + 5 + 4 + 8 = 29 which becomes A + 24 = 29 Subtract 24 from each side of the above equation: A + 24 – 24 = 29 – 24 which makes A = 5 and makes ABCDEF = 570548