Puzzle for June 3, 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 non-negative integer.
* BC is a 2-digit number (not B×C).
Scratchpad
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Hint #1
In eq.4, replace E with B + C (from eq.3): B + B + C = C + F which becomes 2×B + C = C + F Subtract C from both sides of the above equation: 2×B + C - C = C + F - C which makes 2×B = F
Hint #2
In eq.2, replace F with 2×B: 2×B = A - B Add B to both sides of the above equation: 2×B + B = A - B + B which makes 3×B = A
Hint #3
eq.6 may be written as: 10×B + C = D + E + F Substitute B + C for E (from eq.3), and 2×B for F in the above equation: 10×B + C = D + B + C + 2×B which becomes 10×B + C = D + 3×B + C Subtract both 3×B and C from each side of the above equation: 10×B + C - 3×B - C = D + 3×B + C - 3×B - C which simplifies to 7×B = D
Hint #4
Substitute 3×B for A, and 7×B for D in eq.5: 3×B + 7×B = B + E which becomes 10×B = B + E Subtract B from each side of the equation above: 10×B - B = B + E - B which makes 9×B = E
Hint #5
Substitute 9×B for E, and 2×B for F in eq.4: B + 9×B = C + 2×B which becomes 10×B = C + 2×B Subtract 2×B from each side: 10×B - 2×B = C + 2×B - 2×B which makes 8×B = C
Solution
Substitute 3×B for A, 8×B for C, 7×B for D, 9×B for E, and 2×B for F in eq.1: 3×B + B + 8×B + 7×B + 9×B + 2×B = 30 which simplifies to 30×B = 30 Divide both sides by 30: 30×B ÷ 30 = 30 ÷ 30 which means B = 1 making A = 3×B = 3 × 1 = 3 C = 8×B = 8 × 1 = 8 D = 7×B = 7 × 1 = 7 E = 9×B = 9 × 1 = 9 F = 2×B = 2 × 1 = 2 and ABCDEF = 318792