Math Problems, Number Theory Problems

Problem of the Day: 1/15/13

How many four-digit numbers are divisible by 11 and have a sum of digits of 10?

Solution to yesterday’s problem:

Let’s designate the square’s side length a. Then the area of the square is a^2. To find the area of the triangle, we need to find its height. First we’ll draw the altitude to make two 30-60-90 right triangles (whose side lengths are s, s\sqrt{3}, and 2s). The short side of each triangle is \frac{1}{2}a, so the vertical side is \frac{1}{2}a\sqrt{3}. Using the triangle area formula,



Now we can write our two areas as a ratio:


Canceling and multiplying both sides by four, we get the final answer \frac{\sqrt{3}}{4}.


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