Puzzle for July 18, 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.6, replace C with D + E (from eq.4): D + E – F = E + F In the above equation, subtract E from both sides, and add F to both sides: D + E – F – E + F = E + F – E + F which makes D = 2×F
Hint #2
In eq.2, replace D with 2×F: E = 2×F + F which makes E = 3×F
Hint #3
In eq.4, substitute 2×F for D, and 3×F for E: C = 2×F + 3×F which makes C = 5×F
Hint #4
Substitute 5×F for C in eq.5: B – F = 5×F + F which makes B = 7×F
Hint #5
Substitute 7×F for B, and 2×F for D in eq.3: 7×F = A + 2×F Subtract 2×F from both sides of the above equation: 7×F – 2×F = A + 2×F – 2×F which makes 5×F = A
Solution
Substitute 5×F for A and C, 7×F for B, 2×F for D, and 3×F for E in eq.1: 5×F + 7×F + 5×F + 2×F + 3×F + F = 23 which simplifies to 23×F = 23 Divide both sides of the above equation by 23: 23×F ÷ 23 = 23 ÷ 23 which means F = 1 making A = C = 5×F = 5 × 1 = 5 B = 7×F = 7 × 1 = 7 D = 2×F = 2 × 1 = 2 E = 3×F = 3 × 1 = 3 and ABCDEF = 575231