Puzzle for April 25, 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
Subtract the left and right sides of eq.4 from the left and right sides of eq.3, respectively: A + C – (A – C) = E – C – (E – D) which is the same as A + C – A + C = E – C – E + D which becomes 2×C = –C + D Add C to both sides of the equation above: 2×C + C = –C + D + C which makes 3×C = D
Hint #2
In eq.6, replace D with 3×C: F = A + C + 3×C which becomes eq.6a) F = A + 4×C which may also be written as eq.6b) F = A + 2×C + 2×C
Hint #3
Add C to each side of eq.6c: F + C = E + 2×C + C which becomes F + C = E + 3×C In the equation above, substitute D for 3×C: F + C = E + D which may be written as eq.6d) C + F = D + E
Hint #4
Substitute C + F for D + E (from eq.6d) in eq.2: B + C + F = C + F Subtract C and F from both sides of the above equation: B + C + F – C – F = C + F – C – F which simplifies to B = 0
Hint #5
Substitute 3×C for D, A + 4×C for F (from eq.6a), 0 for B, and A + 2×C for E (from eq.3a) in eq.5: 3×C + A + 4×C = A + 0 + A + 2×C which becomes A + 7×C = 2×A + 2×C Subtract A and 2×C from each side of the above equation: A + 7×C – A – 2×C = 2×A + 2×C – A – 2×C which simplifies to 5×C = A
Hint #6
Substitute 5×C for A in eq.3a: 5×C + 2×C = E which makes 7×C = E
Hint #7
Substitute 7×C for E in eq.6c: F = 7×C + 2×C which makes F = 9×C
Solution
Substitute 5×C for A, 0 for B, 3×C for D, 7×C for E, and 9×C for F in eq.1: 5×C + 0 + C + 3×C + 7×C + 9×C = 25 which simplifies to 25×C = 25 Divide both sides of the equation above by 25: 25×C ÷ 25 = 25 ÷ 25 which means C = 1 making A = 5×C = 5 × 1 = 5 D = 3×C = 3 × 1 = 3 E = 7×C = 7 × 1 = 7 F = 9×C = 9 × 1 = 9 and ABCDEF = 501379