Puzzle for June 6, 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.5, replace B with D + F (from eq.2): E + F = D + F – E In the equation above, subtract F from both sides, and add E to both sides: E + F – F + E = D + F – E – F + E which makes eq.2a) 2×E = D
Hint #2
In eq.3, replace D with 2×E (from eq.2a): F = 2×E – E – F which becomes F = E – F Add F to both sides of the above equation: F + F = E – F + F which makes 2×F = E
Hint #3
In eq.2a, substitute (2×F) for E: 2×(2×F) = D which makes 4×F = D
Hint #4
In eq.2, substitute 4×F for D: B = 4×F + F which makes B = 5×F
Hint #5
Substitute 4×F for D, and 2×F for E in eq.4: A – F = 4×F + 2×F which makes A – F = 6×F Add F to both sides of the equation above: A – F + F = 6×F + F which makes A = 7×F
Hint #6
Substitute 2×F for E, 5×F for B, and 4×F for D in eq.6: C + 2×F = 5×F + 4×F which makes C + 2×F = 9×F Subtract F from each side of the equation above: C + 2×F – 2×F = 9×F – 2×F which makes C = 7×F
Solution
Substitute 7×F for A and C, 5×F for B, 4×F for D, 2×F for E in eq.1: 7×F + 5×F + 7×F + 4×F + 2×F + F = 26 which simplifies to 26×F = 26 Divide both sides of the above equation by 26: 26×F ÷ 26 = 26 ÷ 26 which means F = 1 making A = C = 7×F = 7 × 1 = 7 B = 5×F = 5 × 1 = 5 D = 4×F = 4 × 1 = 4 E = 2×F = 2 × 1 = 2 and ABCDEF = 757421