Puzzle for February 25, 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 is a 2-digit number (not A×B).
Scratchpad
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Hint #1
In eq.5, replace C with B + D (from eq.4): A + B + B + D = D + E Subtract D from each side of the above equation: A + B + B + D - D = D + E - D which becomes A + 2×B = E In eq.2, replace E with A + 2×B: B + A + 2×B = A which becomes A + 3×B = A Subtract A from each side: A + 3×B - A = A - A which simplifies to 3×B = 0 which makes B = 0
Hint #2
In eq.2, substitute 0 for B: 0 + E = A which means E = A
Hint #3
In eq.3, substitute A for E: F = A + A which makes F = 2×A
Hint #4
In eq.4, substitute 0 for B: C = 0 + D which means C = D
Hint #5
eq.6 may be written as: 10×A + B = D + E Substitute 0 for B, and A for E in the above equation: 10×A + 0 = D + A Subtract A from each side: 10×A + 0 - A = D + A - A which makes 9×A = D and which means C = D = 9×A
Solution
Substitute 0 for B, 9×A for C and D, A for E, and 2×A for F in eq.1: A + 0 + 9×A + 9×A + A + 2×A = 22 which simplifies to 22×A = 22 Divide both sides of the above equation by 22: 22×A ÷ 22 = 22 ÷ 22 which means A = 1 making C = D = 9×A = 9 × 1 = 9 E = A = 1 F = 2×A = 2 × 1 = 2 and ABCDEF = 109912