Puzzle for January 11, 2021 ( )
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.4, replace E with B + C (from eq.3): C + F = B + B + C which becomes C + F = 2×B + C Subtract C from each side of the above equation: C + F – C = 2×B + C – C which makes F = 2×B
Hint #2
In eq.5, replace F with 2×B: B + 2×B = C – B which becomes 3×B = C – B Add B to both sides of the equation above: 3×B + B = C – B + B which makes 4×B = C
Hint #3
In eq.3, substitute 4×B for C: E = B + 4×B which makes E = 5×B
Hint #4
Substitute 5×B for E, and 2×B for F in eq.6: A = 5×B + 2×B which makes A = 7×B
Hint #5
Substitute 4×B for C, and 5×B for E in eq.2: D = 4×B + 5×B which makes D = 9×B
Solution
Substitute 7×B for A, 4×B for C, 9×B for D, 5×B for E, and 2×B for F in eq.1: 7×B + B + 4×B + 9×B + 5×B + 2×B = 28 which simplifies to 28×B = 28 Divide both sides of the above equation by 28: 28×B ÷ 28 = 28 ÷ 28 which means B = 1 making A = 7×B = 7 × 1 = 7 C = 4×B = 4 × 1 = 4 D = 9×B = 9 × 1 = 9 E = 5×B = 5 × 1 = 5 F = 2×B = 2 × 1 = 2 and ABCDEF = 714952