Puzzle for December 17, 2018 ( )
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.
Scratchpad
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Hint #1
In eq.6, replace F with B + C (from eq.4): A - B + C = B + B + C Subtract C from, and add B to each side of the above equation: A - B + C - C + B = B + B + C - C + B which simplifies to A = 3×B
Hint #2
In eq.2, substitute 3×B for A: 3×B + B = C which means 4×B = C
Hint #3
In eq.4, replace C with 4×B: B + 4×B = F which means 5×B = F
Hint #4
Substitute 4×B for C, and 5×B for F in eq.5: 4×B + 5×B = B + E which becomes 9×B = B + E Subtract B from both sides: 9×B - B = B + E - B which means 8×B = E
Hint #5
Substitute 4×B for C, and 8×B for E in eq.3: 4×B + D = 8×B Subtract 4×B from each side: 4×B + D - 4×B = 8×B - 4×B which simplifies to D = 4×B
Solution
Substitute 3×B for A, 4×B for C and D, 8×B for E, and 5×B for F in eq.1: 3×B + B + 4×B + 4×B + 8×B + 5×B = 25 which becomes 25×B = 25 Divide both sides of the equation above by 25: 25×B ÷ 25 = 25 ÷ 25 which means B = 1 making A = 3×B = 3 × 1 = 3 C = D = 4×B = 4 × 1 = 4 E = 8×B = 8 × 1 = 8 F = 5×B = 5 × 1 = 5 and ABCDEF = 314485