Puzzle for February 7, 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 F with A + E (from eq.2): D + E – A = A + A + E which becomes D + E – A = 2×A + E In the above equation, subtract E from both sides, and add A to both sides: D + E – A – E + A = 2×A + E – E + A which makes D = 3×A
Hint #2
In eq.5, replace D with 3×A: C – 3×A = 3×A – A which becomes C – 3×A = 2×A Add 3×A to both sides of the above equation: C – 3×A + 3×A = 2×A + 3×A which makes C = 5×A
Hint #3
In eq.3, substitute 5×A for C, and 3×A for D: E = 5×A + 3×A which makes E = 8×A
Hint #4
Substitute 8×A for E in eq.2: F = A + 8×A which makes F = 9×A
Hint #5
Substitute 9×A for F, and 3×A for D in eq.4: 9×A = B + 3×A Subtract 3×A from each side of the equation above: 9×A – 3×A = B + 3×A – 3×A which makes 6×A = B
Solution
Substitute 6×A for B, 5×A for C, 3×A for D, 8×A for E, and 9×A for F in eq.1: A + 6×A + 5×A + 3×A + 8×A + 9×A = 32 which simplifies to 32×A = 32 Divide both sides of the above equation by 32: 32×A ÷ 32 = 32 ÷ 32 which means A = 1 making B = 6×A = 6 × 1 = 6 C = 5×A = 5 × 1 = 5 D = 3×A = 3 × 1 = 3 E = 8×A = 8 × 1 = 8 F = 9×A = 9 × 1 = 9 and ABCDEF = 165389