Puzzle for July 25, 2022 ( )
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 + E (from eq.5): E – B = B + E – E which becomes E – B = B Add B to both sides of the above equation: E – B + B = B + B which makes E = 2×B
Hint #2
In eq.5, replace E with 2×B: F = B + 2×B which makes F = 3×B
Hint #3
In eq.4, substitute 2×B for E: 2×B = A + B Subtract B from each side of the above equation: 2×B – B = A + B – B which makes B = A
Hint #4
Substitute B for A, and 3×B for F in eq.3: D = B + 3×B which makes D = 4×B
Hint #5
Substitute 4×B for D, and 3×B for F in eq.2: C = 4×B + 3×B which makes C = 7×B
Solution
Substitute B for A, 7×B for C, 4×B for D, 2×B for E, and 3×B for F in eq.1: B + B + 7×B + 4×B + 2×B + 3×B = 18 which simplifies to 18×B = 18 Divide both sides of the above equation by 18: 18×B ÷ 18 = 18 ÷ 18 which means B = 1 making A = B = 1 C = 7×B = 7 × 1 = 7 D = 4×B = 4 × 1 = 4 E = 2×B = 2 × 1 = 2 F = 3×B = 3 × 1 = 3 and ABCDEF = 117423