Puzzle for January 9, 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.
* CD and EF are 2-digit numbers (not C×D or E×F).
Scratchpad
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Hint #1
Add A to each side of eq.3: C - A + A = A + F + A which becomes eq.3a) C = 2×A + F Subtract 2×A from each side of eq.3a: C - 2×A = 2×A + F - 2×A which becomes eq.3b) C - 2×A = F
Hint #2
In eq.4, replace C with 2×A + F (from eq.3a): E + F = B + 2×A + F - A Subtract F from each side of the above equation: E + F - F = B + 2×A + F - A - F which simplifies to E = B + A In eq.2, replace B + A with E: E = D
Hint #3
In eq.5, substitute D for E: B + D = C + D Subtract D from both sides: B + D - D = C + D - D which means B = C
Hint #4
eq.6 may be written as: 10×E + F - (10×C + D) = E which is the same as 10×E + F - 10×C - D = E Subtract E from each side of the equation above: 10×E + F - 10×C - D - E = E - E which becomes 9×E + F - 10×C - D = 0 Substitute D for E, and C - 2×A for F (from eq.3b): 9×D + C - 2×A - 10×C - D = 0 eq.6a) 8×D - 2×A - 9×C = 0
Hint #5
Subtract B from each side of eq.2: B + A - B = D - B which becomes A = D - B Substitute (D - B) for A, and B for C in eq.6a: 8×D - 2×(D - B) - 9×B = 0 which is the same as 8×D - 2×D + 2×B - 9×B = 0 which becomes 6×D - 7×B = 0 Add 7×B to both sides: 6×D - 7×B + 7×B = 0 + 7×B which means 6×D = 7×B Divide both sides by 6: 6×D ÷ 6 = 7×B ÷ 6 which makes D = 1⅙×B
Hint #6
In eq.2, replace D with 1⅙×B: B + A = 1⅙×B Subtract B from both sides: B + A - B = 1⅙×B - B which makes A = ⅙×B
Hint #7
Substitute (⅙×B) for A, and B for C in eq.3b: B - 2×(⅙×B) = F which becomes B - ⅓×B = F which makes ⅔×B = F
Solution
Substitute ⅙×B for A, B for C, 1⅙×B for D and E, and ⅔×B for F in eq.1: ⅙×B + B + B + 1⅙×B + 1⅙×B + ⅔×B = 31 which simplifies to 5⅙×B = 31 Divide both sides by 5⅙: 5⅙×B ÷ 5⅙ = 31 ÷ 5⅙ which makes B = 6 making A = ⅙×B = ⅙ × 6 = 1 C = B = 6 D = E = 1⅙×B = 1⅙ × 6 = 7 F = ⅔×B = ⅔ × 6 = 4 and ABCDEF = 166774