Puzzle for August 15, 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.
Scratchpad
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Hint #1
In eq.4, substitute (A – B) for F (from eq.2): C = A – (A – B) which becomes C = A – A + B which makes C = B
Hint #2
In eq.6, replace E with D + F (from eq.3): B + D = D + F + F which becomes B + D = D + 2×F Subtract D from each side of the equation above: B + D – D = D + 2×F – D which makes B = 2×F and also makes C = B = 2×F
Hint #3
In eq.2, replace B with 2×F: F = A – 2×F Add 2×F to both sides of the above equation: F + 2×F = A – 2×F – 2×F which makes 3×F = A
Hint #4
In eq.5, substitute 2×F for B and C: D = 2×F + 2×F + F which makes D = 5×F
Hint #5
Substitute 5×F for D in eq.3: E = 5×F + F which makes E = 6×F
Solution
Substitute 3×F for A, 2×F for B and C, 5×F for D, and 6×F for E in eq.1: 3×F + 2×F + 2×F + 5×F + 6×F + F = 19 which simplifies to 19×F = 19 Divide both sides of the above equation by 19: 19×F ÷ 19 = 19 ÷ 19 which means F = 1 making A = 3×F = 3 × 1 = 3 B = C = 2×F = 2 × 1 = 2 D = 5×F = 5 × 1 = 5 E = 6×F = 6 × 1 = 6 and ABCDEF = 322561