Puzzle for February 20, 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 CD are 2-digit numbers (not A×B or C×D).
Scratchpad
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Hint #1
Add (A - D) to both sides of eq.3: E - A + (A - D) = A + D + (A - D) which becomes E - D = 2×A In eq.4, replace E - D with 2×A: eq.4a) C = 2×A + F
Hint #2
In eq.5, substitute (2×A + F) for C (from eq.4a): A - (2×A + F) - F = F - A which is the same as A - 2×A - F - F = F - A which becomes -A - 2×F = F - A Add A + 2×F to each side of the above equation: -A - 2×F + A + 2×F = F - A + A + 2×F which simplifies to 0 = 3×F which means 0 = F
Hint #3
In eq.2, substitute 0 for F: B = D - 0 which means B = D
Hint #4
In eq.4a, substitute 0 for F: C = 2×A + 0 which makes C = 2×A
Hint #5
Substitute B for D, and add A to each side of eq.3: E - A + A = A + B + A which becomes eq.3a) E = 2×A + B
Hint #6
eq.6 may be written as: 10×C + D - (10×A + B) = D + E which is equivalent to 10×C + D - 10×A - B = D + E Add (B - D) to both sides of the equation above: 10×C + D - 10×A - B + (B - D) = D + E + (B - D) which becomes eq.6a) 10×C - 10×A = E + B
Hint #7
Substitute 2×A for C, and 2×A + B for E (from eq.3a) into eq.6a: 10×2×A - 10×A = 2×A + B + B which becomes 20×A - 10×A = 2×A + 2×B which becomes 10×A = 2×A + 2×B Subtract 2×A from both sides of the above equation: 10×A - 2×A = 2×A + 2×B - 2×A which makes 8×A = 2×B Divide both sides by 2: 8×A ÷ 2 = 2×B ÷ 2 which means 4×A = B and also makes D = B = 4×A
Hint #8
Substitute 4×A for B in eq.3a: E = 2×A + 4×A which makes E = 6×A
Solution
Substitute 4×A for B and D, 2×A for C, 6×A for E, and 0 for F in eq.1: A + 4×A + 2×A + 4×A + 6×A + 0 = 17 which becomes 17×A = 17 Divide both sides by 17: 17×A ÷ 17 = 17 ÷ 17 which makes A = 1 making B = D = 4×A = 4 × 1 = 4 C = 2×A = 2 × 1 = 2 E = 6×A = 6 × 1 = 6 and ABCDEF = 142460