Puzzle for November 1, 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
In eq.4, replace D with C + F (from eq.3): C + F – A = A + F In the above equation, subtract F from both sides, and add A to both sides: C + F – A – F + A = A + F – F + A which simplifies to C = 2×A
Hint #2
In eq.5, replace C with 2×A: E = A + 2×A which makes E = 3×A
Hint #3
In eq.6, substitute 2×A for C, and 3×A for E: F = 2×A + 3×A which makes F = 5×A
Hint #4
Substitute 2×A for C, and 5×A for F in eq.3: 2×A + 5×A = D which makes 7×A = D
Hint #5
Substitute 7×A for D into eq.2: B = 7×A – A which makes B = 6×A
Solution
Substitute 6×A for B, 2×A for C, 7×A for D, 3×A for E, and 5×A for F in eq.1: A + 6×A + 2×A + 7×A + 3×A + 5×A = 24 which simplifies to 24×A = 24 Divide both sides of the above equation by 24: 24×A ÷ 24 = 24 ÷ 24 which means A = 1 making B = 6×A = 6 × 1 = 6 C = 2×A = 2 × 1 = 2 D = 7×A = 7 × 1 = 7 E = 3×A = 3 × 1 = 3 F = 5×A = 5 × 1 = 5 and ABCDEF = 162735