Puzzle for June 19, 2022 ( )
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.
* "A mod B" equals the remainder of (A ÷ B).
Scratchpad
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Hint #1
In eq.5, add B to both sides, and subtract (D ÷ F) from both sides: E + (D ÷ F) + B – (D ÷ F) = (C ÷ F) – B + B – (D ÷ F) which becomes E + B = (C ÷ F) – (D ÷ F) which is equivalent to eq.5a) B + E = (C – D) ÷ F
Hint #2
In eq.5a, replace C with D + F (from eq.2): B + E = (D + F – D) ÷ F which becomes B + E = (F) ÷ F which becomes eq.5b) B + E = 1
Hint #3
Since B and E are one-digit non-negative integers, then eq.5b makes: B = 1 and E = 0 or B = 0 and E = 1 Since B ≠ 0 (from eq.6), then: B = 1 and E = 0
Hint #4
eq.4 may be written as: A = (B + C + E + F) ÷ 4 Multiply both sides of the above equation by 4: 4 × A = 4 × (B + C + E + F) ÷ 4 which becomes eq.4a) 4×A = B + C + E + F
Hint #5
eq.1 may be re-written as: B + C + E + F + A + D = 18 In the above equation, replace B + C + E + F with 4×A (from eq.4a): 4×A + A + D = 18 which becomes 5×A + D = 18 Subtract 5×A from each side: 5×A + D – 5×A = 18 – 5×A which becomes eq.1a) D = 18 – 5×A
Hint #6
eq.1 may also be re-written as: A + B + C + E + D + F = 18 In the equation above, substitute 1 for B, 0 for E, and C for D + F (from eq.2): A + 1 + C + 0 + C = 18 which becomes A + 1 + 2×C = 18 Subtract 1 from each side: A + 1 + 2×C – 1 = 18 – 1 which becomes eq.1b) A + 2×C = 17
Hint #7
Substitute 1 for B, and 0 for E in eq.3: C – 1 = A + D – 0 which becomes C – 1 = A + D Add 1 to both sides of the equation above: C – 1 + 1 = A + D + 1 which becomes eq.3a) C = A + D + 1
Hint #8
Substitute (A + D + 1) for C (from eq.3a) into eq.1b: A + 2×(A + D + 1) = 17 which becomes A + 2×A + 2×D + 2×1 = 17 which becomes 3×A + 2×D + 2 = 17 Subtract 2 from each side of the above equation: 3×A + 2×D + 2 – 2 = 17 – 2 which becomes eq.1c) 3×A + 2×D = 15
Hint #9
Substitute (18 – 5×A) for D (from eq.1a) in eq.1c: 3×A + 2×(18 – 5×A) = 15 which becomes 3×A + 36 – 10×A = 15 which becomes 36 – 7×A = 15 Subtract 36 from both sides of the equation above: 36 – 7×A – 36 = 15 – 36 which becomes –7×A = –21 Divide both sides by (–7): –7×A ÷ (–7) = –21 ÷ (–7) which makes A = 3
Hint #10
Substitute 3 for A in eq.1a: D = 18 – 5×3 which becomes D = 18 – 15 which makes D = 3
Hint #11
Substitute 3 for A and D in eq.3a: C = 3 + 3 + 1 which makes C = 7
Solution
Substitute 7 for C, and 3 for D in eq.2: 7 = 3 + F Subtract 3 from each side of the above equation: 7 – 3 = 3 + F – 3 which makes 4 = F and makes ABCDEF = 317304