Puzzle for September 21, 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.3, replace E with A + D (from eq.4): D + A + D = A + B which becomes A + 2×D = A + B Subtract A from each side of the above equation: A + 2×D – A = A + B – A which makes eq.3a) 2×D = B
Hint #2
In eq.6, replace B with 2×D: 2×D – A = A + D In the above equation, add A to both sides, and subtract D from both sides: 2×D – A + A – D = A + D + A – D which makes D = 2×A
Hint #3
In eq.3a, replace D with (2×A): 2×(2×A) = B which makes 4×A = B
Hint #4
Substitute 2×A for D, and 4×A for B in eq.3: 2×A + E = A + 4×A which becomes 2×A + E = 5×A Subtract 2×A from both sides of the equation above: 2×A + E – 2×A = 5×A – 2×A which makes E = 3×A
Hint #5
Substitute 4×A for B, and 3×A for E in eq.5: C = 4×A + 3×A which makes C = 7×A
Hint #6
Substitute 7×A for C, and 2×A for D in eq.2: 7×A = 2×A + F Subtract 2×A from each side of the equation above: 7×A – 2×A = 2×A + F – 2×A which makes 5×A = F
Solution
Substitute 4×A for B, 7×A for C, 2×A for D, 3×A for E, and 5×A for F in eq.1: A + 4×A + 7×A + 2×A + 3×A + 5×A = 22 which simplifies to 22×A = 22 Divide both sides of the equation above by 22: 22×A ÷ 22 = 22 ÷ 22 which means A = 1 making B = 4×A = 4 × 1 = 4 C = 7×A = 7 × 1 = 7 D = 2×A = 2 × 1 = 2 E = 3×A = 3 × 1 = 3 F = 5×A = 5 × 1 = 5 and ABCDEF = 147235