Puzzle for May 4, 2020 ( )
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 B with C + D (from eq.3): C + D – A = A + C In the equation above, subtract C from both sides, and add A to both sides: C + D – A – C + A = A + C – C + A which makes D = 2×A
Hint #2
In eq.6, replace D with 2×A: F – A = A + 2×A which becomes F – A = 3×A Add A to both sides of the equation above: F – A + A = 3×A + A which makes F = 4×A
Hint #3
In eq.5, substitute 2×A for D, and 4×A for F: 2×A = C – 4×A Add 4×A to both sides of the above equation: 2×A + 4×A = C – 4×A + 4×A which makes 6×A = C
Hint #4
Substitute 6×A for C, and 2×A for D in eq.3: 6×A + 2×A = B which makes 8×A = B
Hint #5
Substitute 8×A for B in eq.2: E = A + 8×A which makes E = 9×A
Solution
Substitute 8×A for B, 6×A for C, 2×A for D, 9×A for E, and 4×A for F in eq.1: A + 8×A + 6×A + 2×A + 9×A + 4×A = 30 which simplifies to 30×A = 30 Divide both sides of the equation above by 30: 30×A ÷ 30 = 30 ÷ 30 which means A = 1 making B = 8×A = 8×1 = 8 C = 6×A = 6×1 = 6 D = 2×A = 2×1 = 2 E = 9×A = 9×1 = 9 F = 4×A = 4×1 = 4 and ABCDEF = 186294