Puzzle for May 16, 2019 ( )
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.
* AB and EF are 2-digit numbers (not A×B or E×F).
Scratchpad
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Hint #1
Add A, B, and F to both sides of eq.4: C - F + A + B + F = E - A - B + A + B + F which becomes C + A + B = E + F which may be written as eq.4a) A + B + C = E + F In eq.2, replace A + B + C with E + F (from eq.4a): E + F = D + F Subtract F from both sides of the above equation: E + F - F = D + F - F which means E = D
Hint #2
In eq.1, replace D with E, and replace A + B + C with E + F (from eq.4a): E + F + E + E + F = 45 which becomes 3×E + 2×F = 45 Subtract 3×E from both sides of the above equation: 3×E + 2×F - 3×E = 45 - 3×E which becomes eq.1a) 2×F = 45 - 3×E
Hint #3
To make eq.1a true, check several possible values for E and F: If E = 9, then 2×F = 45 - 3×9 = 45 - 27 = 18 making F = 18 ÷ 2 = 9 If E = 8, then 2×F = 45 - 3×8 = 45 - 24 = 21 making F = 21 ÷ 2 = 10½ If E < 8, then F > 10½ Since F must be a one-digit integer, then F = 9 which makes E = 9 which also makes D = E = 9
Hint #4
In eq.5, replace E with D: B + D = A - B + D In the equation above, subtract D from both sides, and add B to both sides: B + D - D + B = A - B + D - D + B which makes 2×B = A
Hint #5
Substitute 2×B for A, and 9 for D and F in eq.2: 2×B + B + C = 9 + 9 which becomes 3×B + C = 18 Subtract 3×B from both sides: 3×B + C - 3×B = 18 - 3×B which becomes eq.2a) C = 18 - 3×B
Solution
eq.3 may be written as: 10×E + F = 10×A + B + C + D Substitute 2×B for A, 18 - 3×B for C (from eq.2a), and 9 for D and E and F in the equation above: 10×9 + 9 = 10×2×B + B + 18 - 3×B + 9 which becomes 99 = 20×B + B - 3×B + 27 which becomes 99 = 18×B + 27 Subtract 27 from each side: 99 - 27 = 18×B + 27 - 27 which makes 72 = 18×B Divide both sides by 18: 72 ÷ 18 = 18×B ÷ 18 which means 4 = B making A = 2×B = 2 × 4 = 8 C = 18 - 3×B = 18 - 3 × 4 = 18 - 12 = 6 (from eq.2a) and ABCDEF = 846999