<p>The figure above represents a right circular cylinder made of paper. It has a height of 20 centimeters and a diameter of 8 centimeters, and it is open at both ends. Of the following, which is closest to the surface area, in square centimeters, of the outside of the paper cylinder?</p>
<p>(A)450
(B)475
(C) 500
(D) 525
(E) 550</p>
<p>The integer n is equal to k2 for some integer k. If n is divisible by 24 and by 10, what is the smallest possible positive value of n?</p>
<p>I got it right but I used a long method which took too much time. What is the fast way to do this?</p>
<p>im thinking the first one is 8<em>pi</em>20 but why would need to multiply by pi for the surface area. its just a rectangle without the top and bottom</p>
<p>1) imagine the cylinder as a rectangle. the area of the rectangle = base<em>height
the length of the base of the cylinder is simply its circumference, which is equal to 8</em>pi</p>
<p>So, 20<em>8</em>pi =502.4 or 500 so the answer is C</p>
<ol>
<li>n must be divisible by LCM(24,10), or 120.</li>
</ol>
<p>Note that 120 = (2^3)(3)(5), and n = k^2 is divisible by 120. So we want the smallest multiple of 120 that is a perfect square. If we multiply by 2<em>3</em>5, all the exponents will be even, hence perfect square. Therefore n = 120<em>2</em>3*5 = 3600.</p>