Puzzle for September 12, 2019 ( )
Scratchpad
Find the 5-digit number ABCDE by solving the following equations:
A, B, C, D, and E each represent a one-digit non-negative integer.
Scratchpad
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Hint #1
eq.4 may be written as: B – D + E = A + D + E Subtract E from each side of the above equation: B – D + E – E = A + D + E – E which becomes eq.4a) B – D = A + D
Hint #2
eq.3 may be written as: B – D = A + D + C In the above equation, replace A + D with B – D (from eq.4a): B – D = B – D + C Add D to each side, and subtract B from each side: B – D + D – B = B – D + C + D – B which makes 0 = C
Hint #3
In eq.2, replace C with 0: A = 0 + D which means A = D
Hint #4
Substitute D for A, and 0 for C in eq.3: B – D = D + 0 + D which becomes B – D = 2×D Add D to both sides: B – D + D = 2×D + D which makes B = 3×D
Hint #5
Substitute D for A, and 3×D for B in eq.5: eq.5a) D × D = 3×D which means D = 0 or D = 3
Hint #6
To make eq.5a true, check the possible values for D: Check D = 0 ... If D = 0, then A = D = 0 and B = 3×0 = 0 Substituting 0 for A, B, C, and D in eq.1 would yield 0 + 0 + 0 + 0 + E = 16 which would mean E = 16 Since E is a one-digit integer, then E ≠ 16 which means D ≠ 0 and therefore means D = 3 making A = D = 3 B = 3×D = 3×3 = 9
Solution
Substitute 3 for A and D, 9 for B, and 0 for C in eq.1: 3 + 9 + 0 + 3 + E = 16 which simplifies to 15 + E = 16 Subtract 15 from both sides of the equation above: 15 + E – 15 = 16 – 15 which means E = 1 making ABCDE = 39031