Puzzle for March 5, 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.
* BC is a 2-digit number (not B×C).
Scratchpad
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Hint #1
In eq.4, replace D with B + F (from eq.3): B + F - A = B + A Add (A - B) to both sides of the above equation: B + F - A + (A - B) = B + A + (A - B) which makes eq.4a) F = 2×A
Hint #2
In eq.5, replace F with 2×A (from eq.4a): 2×A - A = C + A Subtract A from each side of the above equation: 2×A - A - A = C + A - A which makes 0 = C
Hint #3
In eq.2, substitute 0 for C: 0 + D = E which means D = E
Hint #4
eq.6 may be written as: 10×B + C + F = D + E Substitute 0 for C, and D for E in the equation above: 10×B + 0 + F = D + D which becomes eq.6a) 10×B + F = 2×D
Hint #5
Substitute (B + F) for D (from eq.3) in eq.6a: 10×B + F = 2×(B + F) which is equivalent to 10×B + F = 2×B + 2×F Subtract both F and 2×B from each side of the above equation: 10×B + F - F - 2×B = 2×B + 2×F - F - 2×B which simplifies to 8×B = F
Hint #6
Substitute 8×B for F in eq.4a: 8×B = 2×A Divide both sides of the equation above by 2: 8×B ÷ 2 = 2×A ÷ 2 which makes 4×B = A
Hint #7
Substitute 4×B for A in eq.4: D - 4×B = B + 4×B Add 4×B to each side: D - 4×B + 4×B = B + 4×B + 4×B which makes D = 9×B which also makes E = D = 9×B
Solution
Substitute 4×B for A, 0 for C, 9×B for D and E, and 8×B for F in eq.1: 4×B + B + 0 + 9×B + 9×B + 8×B = 31 which simplifies to 31×B = 31 Divide both sides of the above equation by 31: 31×B ÷ 31 = 31 ÷ 31 which means B = 1 making A = 4×B = 4 × 1 = 4 D = E = 9×B = 9 × 1 = 9 F = 8×B = 8 × 1 = 8 and ABCDEF = 410998