Puzzle for June 9, 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.
Scratchpad
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Hint #1
Subtract E from both sides of eq.3: A + B – E = C + E – E which becomes A + B – E = C In the equation above, replace B – E with F (from eq.2): eq.3a) A + F = C
Hint #2
In eq.4, replace C with A + F (from eq.3a): A + F – E = E – F In the equation above, subtract F from both sides, and add E to both sides: A + F – E – F + E = E – F – F + E which becoems eq.4a) A = 2×E – 2×F
Hint #3
In eq.5, substitute 2×E – 2×F for A (from eq.4a): E + F = 2×E – 2×F – F which becomes E + F = 2×E – 3×F In the above equation, subtract E from both sides, and add 3×F to both sides: E + F – E + 3×F = 2×E – 3×F – E + 3×F which makes 4×F = E
Hint #4
Substitute (4×F) for E in eq.4a: A = 2×(4×F) – 2×F which becomes A = 8×F – 2×F which makes A = 6×F
Hint #5
Substitute 6×F for A in eq.3a: 6×F + F = C which makes 7×F = C
Hint #6
Substitute 4×F for E in eq.2: F = B – 4×F Add 4×F to both sides of the equation above: F + 4×F = B – 4×F + 4×F which makes 5×F = B
Hint #7
Substitute 7×F for C, 4×F for E, 6×F for A, and 5×F for B in eq.6: 7×F + D – 4×F – F = 6×F – 5×F + F which becomes 2×F + D = 2×F Subtract 2×F from each side of the above equation: 2×F + D – 2×F = 2×F – 2×F which makes D = 0
Solution
Substitute 6×F for A, 5×F for B, 7×F for C, 0 for D, and 4×F for E in eq.1: 6×F + 5×F + 7×F + 0 + 4×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 = 6×F = 6 × 1 = 6 B = 5×F = 5 × 1 = 5 C = 7×F = 7 × 1 = 7 E = 4×F = 4 × 1 = 4 and ABCDEF = 657041