Puzzle for July 20, 2020 ( )
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
Add B to both sides of eq.3: B + D + B = F – B + B which becomes eq.3a) 2×B + D = F In eq.4, replace F with 2×B + D (from eq.3a): D – B = 2×B + D – D which becomes D – B = 2×B Add B to both sides of the equation above: D – B + B = 2×B + B which makes D = 3×B
Hint #2
In eq.3a, replace D with 3×B: 2×B + 3×B = F which makes 5×B = F
Hint #3
In eq.2, substitute 3×B for D, and 5×B for F: 3×B + 5×B = E which makes 8×B = E
Hint #4
Substitute 8×B for E in eq.6: B = C – 8×B Add 8×B to both sides of the above equation: B + 8×B = C – 8×B + 8×B which makes 9×B = C
Hint #5
Substitute 3×B for D, and 9×B for C in eq.5: A – B + 3×B = B + 9×B which becomes A + 2×B = 10×B Subtract 2×B from both sides of the equation above: A + 2×B – 2×B = 10×B – 2×B which makes A = 8×B
Solution
Substitute 8×B for A and E, 9×B for C, 3×B for D, and 5×B for F in eq.1: 8×B + B + 9×B + 3×B + 8×B + 5×B = 34 which simplifies to 34×B = 34 Divide both sides of the equation above by 34: 34×B ÷ 34 = 34 ÷ 34 which means B = 1 making A = E = 8×B = 8 × 1 = 8 C = 9×B = 9 × 1 = 9 D = 3×B = 3 × 1 = 3 F = 5×B = 5 × 1 = 5 and ABCDEF = 819385