Puzzle for October 19, 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.6, replace B with C + E (from eq.4): C + E + E = C + D which becomes C + 2×E = C + D Subtract C from both sides of the above equation: C + 2×E – C = C + D – C which makes eq.6a) 2×E = D
Hint #2
In eq.3, replace D with 2×E, and replace C with A + E (from eq.5): 2×E = A + A + E which becomes 2×E = 2×A + E Subtract E from each side of the above equation: 2×E – E = 2×A + E – E which makes E = 2×A
Hint #3
In eq.6a, substitute (2×A) for E: 2×(2×A) = D which makes 4×A = D
Hint #4
Substitute 4×A for D in eq.3: 4×A = A + C Subtract A from each side of the equation above: 4×A – A = A + C – A which makes 3×A = C
Hint #5
Substitute 3×A for C, and 2×A for E in eq.4: 3×A + 2×A = B which makes 5×A = B
Hint #6
Substitute 5×A for B, and 3×A for C in eq.2: F = 5×A + 3×A which makes F = 8×A
Solution
Substitute 5×A for B, 3×A for C, 4×A for D, 2×A for E, and 8×A for F in eq.1: A + 5×A + 3×A + 4×A + 2×A + 8×A = 23 which simplifies to 23×A = 23 Divide both sides of the above equation by 23: 23×A ÷ 23 = 23 ÷ 23 which means A = 1 making B = 5×A = 5 × A = 5 C = 3×A = 3 × A = 3 D = 4×A = 4 × A = 4 E = 2×A = 2 × A = 2 F = 8×A = 8 × A = 8 and ABCDEF = 153428