Puzzle for March 26, 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, BC, and EF are 2-digit numbers (not A×B, B×C, or E×F).
Scratchpad
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Hint #1
eq.5 may be written as: 10×E + F = D + E + F Subtract both E and F from each side of the equation above: 10×E + F - E - F = D + E + F - E - F which becomes 9×E = D
Hint #2
In eq.3, replace D with 9×E: C + 9×E = E Subtract E from each side: C + 9×E - E = E - E which means C + 9×E = 0 Since C and E must be non-negative integers, the above equation makes C = 0 and E = 0 and which also makes D = 9×E = 9 × 0 = 0
Hint #3
eq.4 may be written as: 10×B + C - B = 10×A + B - D which becomes 9×B + C = 10×A + B - D In the above equation, substitute 0 for both C and D, and subtract B from each side: 9×B + 0 - B = 10×A + B - 0 - B which becomes 8×B = 10×A Divide both sides by 8: 8×B ÷ 8 = 10×A ÷ 8 which makes B = 1¼×A
Hint #4
Substitute 1¼×A for B, and 0 for E in eq.2: 1¼×A - A = F - 0 - 1¼×A which becomes ¼×A = F - 1¼×A Add 1¼×A to each side: ¼×A + 1¼×A = F - 1¼×A + 1¼×A which makes 1½×A = F
Solution
Substitute 1¼×A for B, 0 for C and D and E, and 1½×A for F in eq.1: A + 1¼×A + 0 + 0 + 0 + 1½×A = 15 which simplifies to 3¾×A = 15 Divide both sides by 3¾: 3¾×A ÷ 3¾ = 15 ÷ 3¾ which means A = 4 making B = 1¼×A = 1¼ × 4 = 5 F = 1½×A = 1½ × 4 = 6 and ABCDEF = 450006