Puzzle for June 28, 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.6, replace A with B + D (from eq.3): D + E = B + D + B which becomes D + E = 2×B + D Subtract D from each side of the above equation: D + E – D = 2×B + D – D which makes E = 2×B
Hint #2
In eq.2, replace E with 2×B: F = B + 2×B which makes F = 3×B
Hint #3
In eq.5, substitute 2×B for E, and 3×B for F: C = B + 2×B + 3×B which makes C = 6×B
Hint #4
Substitute 6×B for C, and 3×B for F in eq.4: A + B = 6×B + 3×B which becomes A + B = 9×B Subtract B from both sides of the above equation: A + B – B = 9×B – B which makes A = 8×B
Hint #5
Substitute 8×B for A into eq.3: B + D = 8×B Subtract B from each side of the equation above: B + D – B = 8×B – B which makes D = 7×B
Solution
Substitute 8×B for A, 6×B for C, 7×B for D, 2×B for E, and 3×B for F in eq.1: 8×B + B + 6×B + 7×B + 2×B + 3×B = 27 which simplifies to 27×B = 27 Divide both sides of the above equation by 27: 27×B ÷ 27 = 27 ÷ 27 which means B = 1 making A = 8×B = 8 × 1 = 8 C = 6×B = 6 × 1 = 6 D = 7×B = 7 × 1 = 7 E = 2×B = 2 × 1 = 2 F = 3×B = 3 × 1 = 3 and ABCDEF = 816723