Puzzle for July 2, 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 positive integer.
Scratchpad
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Hint #1
In eq.2, replace A with B + D + E (from eq.6): B + D + E + C = B + D + F Subtract B and D from both sides of the above equation: B + D + E + C - B - D = B + D + F - B - D which becomes E + C = F which may be written as eq.2a) C + E = F
Hint #2
In eq.3, replace C + E with F (from eq.2a): D = F
Hint #3
Add B to both sides of eq.5: F - B + B = B + C + B which becomes F = 2×B + C In the above equation, replace F with C + E (from eq.2a): C + E = 2×B + C Subtract C from ech side: C + E - C = 2×B + C - C which makes E = 2×B
Hint #4
Substitute 2×B for E in eq.4: C = B + 2×B which makes C = 3×B
Hint #5
Substitute 3×B for C, and 2×B for E in eq.3: D = 3×B + 2×B which makes D = 5×B which also makes F = D = 5×B
Hint #6
Substitute 5×B for D, and 2×B for E in eq.6: B + 5×B + 2×B = A which makes 8×B = A
Solution
Substitute 8×B for A, 3×B for C, 5×B for D and F, and 2×B for E in eq.1: 8×B + B + 3×B + 5×B + 2×B + 5×B = 24 which simplifies to 24×B = 24 Divide both sides by 24: 24×B ÷ 24 = 24 ÷ 24 which means B = 1 making A = 8×B = 8 × 1 = 1 C = 3×B = 3 × 1 = 1 D = F = 5×B = 5 × 1 = 1 E = 2×B = 2 × 1 = 1 and ABCDEF = 813525