Puzzle for May 14, 2019 ( )
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.2: E + F - E = A + B - E which becomes F = A + B - E In the equation above, replace B - E with A (from eq.4): F = A + A which means F = 2×A
Hint #2
In eq.5, add D to both sides, and subtract F from both sides: F + D - F = C - D + D - F which becomes D = C - F In the above equation, substitute 2×A for F: eq.5a) D = C - 2×A
Hint #3
Substitute C - 2×A for D (from eq.5a), and 2×A for F in eq.3: C - 2×A + E = C + 2×A Add 2×A to each side, and subtract C from each side: C - 2×A + E + 2×A - C = C + 2×A + 2×A - C which simplifies to E = 4×A
Hint #4
Substitute 4×A for E, and 2×A for F in eq.2: 4×A + 2×A = A + B Subtract A from each side of the equation above: 4×A + 2×A - A = A + B - A which means 5×A = B
Hint #5
Substitute 5×A for B, and 4×A for E in eq.6: 5×A = C - 4×A Add 4×A to both sides: 5×A + 4×A = C - 4×A + 4×A which makes 9×A = C
Hint #6
Substitute 2×A for F, and 9×A for C in eq.5: 2×A = 9×A - D In the above equation, add D to each side, and subtract 2×A from each side: 2×A + D - 2×A = 9×A - D + D - 2×A which makes D = 7×A
Solution
Substitute 5×A for B, 9×A for C, 7×A for D, 4×A for E, and 2×A for F in eq.1: A + 5×A + 9×A + 7×A + 4×A + 2×A = 28 which simplifies to 28×A = 28 Divide both sides by 28: 28×A ÷ 28 = 28 ÷ 28 which means A = 1 making B = 5×A = 5 × 1 = 5 C = 9×A = 9 × 1 = 9 D = 7×A = 7 × 1 = 7 E = 4×A = 4 × 1 = 4 F = 2×A = 2 × 1 = 2 and ABCDEF = 159742