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For any three natural numbers a, b and c, if bc+1 11 abc + a+c 43' then the value of abc is
Explanation
The problem involves solving a system of equations for natural numbers a, b, and c. The given expression is (bc + 1) / (abc + a + c) = 11 / 43. By taking the reciprocal, we get (abc + a + c) / (bc + 1) = 43 / 11. This can be simplified by dividing the numerator by the denominator: [a(bc + 1) + c] / (bc + 1) = a + [c / (bc + 1)] = 43 / 11. Since a, b, and c are natural numbers, we express 43/11 as a mixed fraction: 3 + 10/11. Comparing the parts, we find a = 3 and c / (bc + 1) = 10 / 11. Taking the reciprocal again, (bc + 1) / c = 11 / 10, which simplifies to b + 1/c = 1 + 1/10. This yields b = 1 and c = 10. Thus, the product abc = 3 * 1 * 10 = 30. This method utilizes algebraic manipulation and properties of natural numbers [1].
Sources
- [1] https://www.mathcentre.ac.uk/resources/uploaded/hobsonajjustthemaths20021296smcetp.pdf