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A 3-digit number of different digits is formed out of the digits 1, 3, 5 and 7; a single digit is left over. If we multiply the 3-digit number by the left over single digit, find the greatest and smallest product that we can have.
Study the given figure. $$\ell_1$$ and $$\ell_2$$ are two parallel lines. Two equilateral triangles ABC and DEF are placed as shown in the figure, such that the angle between BC and FD is $$39^\circ$$. Find the angle made by the side ED with the common perpendicular to these parallel lines.
Given $$A = 5 + \cfrac{1}{4+\cfrac{1}{3+\cfrac{1}{2+\cfrac{1}{1+\frac{1}{5}}}}}$$. When simplified, A takes the form $$\frac{p}{q}$$ where p, q are positive integers with no common factors. Prove that $$5\frac{1}{5} < A < 5\frac{2}{5}$$.
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There are two fractions $$f_1$$ and $$f_2$$, both in their lowest terms. The numerator of $$f_2$$ is 3 more than the numerator of $$f_1$$. The denominator of $$f_1$$ is 20 less than the denominator of $$f_2$$. In $$f_1$$, if we subtract twice the numerator from the denominator, we get 3. In $$f_2$$, if three times the numerator is subtracted from the denominator, we get 2. Find the value of $$47f_2 - 27f_1$$.
In the addition sum shown, A, B and C stand for different digits. What is the value of the product ABC?
There are 3 positive numbers a, b and c. The ratio of the first to the sum of the other two is 1 : 3.
The ratio of the third to the sum of the other two is 5 : 7.
Find the ratio of the second to the sum of the other two.
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Find all 5-digit numbers of the form $$34a5b$$ where a, b are digits, such that they are divisible by 36.
For any natural number n, prove that $$\frac{n}{3}$$ + $$\frac{n^2}{2}$$ + $$\frac{n^3}{6}$$ is also a natural number.
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