Puzzle for March 1, 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, CD, and EF are 2-digit numbers (not A×B, B×C, C×D, or E×F).
Scratchpad
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Hint #1
Multiply both sides of eq.2 by 10: 10×(B + C) = 10×E which becomes 10×B + 10×C = 10×E which may also be written as eq.2a) 10×E = 10×B + 10×C
Hint #2
eq.6 may be written as: 10×C + D = 10×E + F - (10×B + C) + C which is equivalent to 10×C + D = 10×E + F - 10×B - C + C which becomes 10×C + D = 10×E + F - 10×B Add the left and right sides of eq.2a to the left and right sides of the above equation, respectively: 10×C + D + 10×E = 10×E + F - 10×B + 10×B + 10×C which becomes 10×C + D + 10×E = 10×E + F + 10×C Subtract both 10×C and 10×E from each side: 10×C + D + 10×E - 10×C - 10×E = 10×E + F + 10×C - 10×C - 10×E which simplifies to D = F
Hint #3
eq.3 may be written as: E + C - A - B = A - E + B In the equation above, replace A - E with B + E - A (from eq.4): E + C - A - B = B + E - A + B which becomes E + C - A - B = 2×B + E - A Add (A + B - E) to both sides: E + C - A - B + (A + B - E) = 2×B + E - A + (A + B - E) which simplifies to C = 3×B
Hint #4
In eq.2, replace C with 3×B: B + 3×B = E which makes 4×B = E
Hint #5
In eq.4, replace E with 4×B: A - 4×B = B + 4×B - A Add 4×B + A to both sides of the equation above: A - 4×B + 4×B + A = B + 4×B - A + 4×B + A which makes 2×A = 9×B Divide both sides by 2: 2×A ÷ 2 = 9×B ÷ 2 which means A = 4½×B
Hint #6
eq.5 may be written as: 10×A + B - C = 10×E + F + C Substitute (4½×B) for A, 3×B for C, and (4×B) for E in this above equation: 10×(4½×B) + B - 3×B = 10×(4×B) + F + 3×B which becomes 45×B + B - 3×B = 40×B + F + 3×B which becomes 43×B = 43×B + F Subtract 43×B from each side: 43×B - 43×B = 43×B + F - 43×B which means 0 = F which also means D = F = 0
Solution
Substitute 4½×B for A, 3×B for C, 0 for D and F, and 4×B for E in eq.1: 4½×B + B + 3×B + 0 + 4×B + 0 = 25 which simplifies to 12½×B = 25 Divide both sides by 12½: 12½×B ÷ 12½ = 25 ÷ 12½ which means B = 2 making A = 4½×B = 4½ × 2 = 9 C = 3×B = 3 × 2 = 6 E = 4×B = 4 × 2 = 8 and ABCDEF = 926080