Puzzle for January 27, 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.2, replace F with A + E (from eq.6), and replace C + D with E (from eq.5): B + A + E = A + E Subtract both A and E from each side of the equation above: B + A + E – A – E = A + E – A – E which simplifies to B = 0
Hint #2
In eq.3, replace F with A + E (from eq.6): C + A + E = A – C + D + E Subtract A and E from both sides of the above equation: C + A + E – A – E = A – C + D + E – A – E which becomes C = –C + D Add C to both sides: C + C = –C + D + C which makes 2×C = D
Hint #3
In eq.5, substitute 2×C for D: E = C + 2×C which makes E = 3×C
Hint #4
Substitute 2×C for D, 3×C for E, and 0 for B in eq.4: 2×C + 3×C = A + 0 which makes 5×C = A
Hint #5
Substitute 5×C for A, and 3×C for E in eq.6: F = 5×C + 3×C which makes F = 8×C
Solution
Substitute 5×C for A, 0 for B, 2×C for D, 3×C for E, and 8×C for F in eq.1: 5×C + 0 + C + 2×C + 3×C + 8×C = 19 which simplifies to 19×C = 19 Divide both sides of the equation above by 19: 19×C ÷ 19 = 19 ÷ 19 which means C = 1 making A = 5×C = 5 × 1 = 5 D = 2×C = 2 × 1 = 2 E = 3×C = 3 × 1 = 3 F = 8×C = 8 × 1 = 8 and ABCDEF = 501238