<p>^ Pretty much.</p>
<p>Level 3 Number Theory</p>
<ol>
<li>What is the largest positive integer value of k for which 5^k divides 75^3?</li>
</ol>
<p>(A) 2
(B) 4
© 6
(D) 7
(E) 8</p>
<p>75^3=(3<em>25)^2=(5^2</em>3)^3=5^6*3^3. Thus 5^6 divides 75^3 and no larger power of 5 does. ©.</p>
<p>That is essentially correct except for a few typos. Can someone fix up the small errors?</p>
<p>And is there any other way to do it?</p>
<p>Sorry, the ^2 in (3*25)^2 should be a ^3. Maybe I’m blind, but what other typos were there?</p>
<p>That was it actually. I thought there was one more here: (5^2*3)^3</p>
<p>but you just interchanged the 25 and the 3, which is okay.</p>
<p>A less insightful approach: use your calculator :)</p>
<p>This would also have been a good “grid-in”. </p>
<p>Pckeller’s calculator method is what I would recommend for most students. Since the word “largest” appears in the answer choices, I would start with choice (E) (the largest number) as my first guess.</p>
<p>Random’s method is very good, but I do not expect the average student to be able to do this.</p>
<p>Level 5 Geometry</p>
<ol>
<li>Consider a right circular cylinder with base diameter 6 and height 9. If point O is the center of the top of the cylinder and B lies on the circumference of the bottom of the cylinder, what is the straight-line distance between O and B?</li>
</ol>
<p>I’m probably missing something since it’s a level 5 but…</p>
<p>draw a line from O to the center of the bottom face. This has length 9. Draw a radius from this center to B. Since the top and bottom faces are congruent, this radius has length 3. These two lines are perpendicular since the axis of a cylinder is perpendicular to the plane of either of the faces. We have a right triangle with legs 3 and 9 whose hypotenuse is the distance between O and B. OB^2=3^2+9^2. OB=√(90)=3√(10)</p>
<p>Yes that’s correct (except you need to get a decimal approximation you can actually grid in). If you think that problem is easy, then you’re in pretty good shape with geometry.</p>
<p>Oh, I’m not doing this for SAT practice; I’m just trying to keep my mind sharp this summer before college. But I do recall at least one (substantially) harder geometry problem from when I did take the SAT.</p>
<p>Dr. S I don’t understand the following: “What is the largest positive integer value of k for which 5^k divides 75^3?” I’m not familiar with the word divides used in this way. Can you explain?</p>
<p>@CHD2013 The formal definition of “divides” is as follows:</p>
<p>An integer m divides an integer n if there is another integer k such that n = mk. </p>
<p>For example, 2 divides 6 because 6 = 2*3.</p>
<p>In practice you can use your calculator to see if one integer divides another by just performing the division and seeing if you get an integer answer. </p>
<p>For example, 2 does not divide 7 because when we divide 7 by 2 we get 3.5, and 3.5 is not an integer.</p>
<p>Hope that helps!</p>
<p>Thanks. Do you know if the word is used that way on the SAT?</p>
<p>The question I just gave you is very similar to a College Board question that was used as a “question of the day.” In particular, the word “divides” was used in that question. So I would say that the word could definitely be used this way on an actual SAT.</p>
<p>Ok, thanks Doc</p>
<p>Level 5 Number Theory</p>
<ol>
<li>If n is a positive integer such that the units (ones) digit of n^2+4n is 7 and the units digit of n is not 7, what is the units digit of n + 3?</li>
</ol>
<p>If the units digit of n^2+4n is 7, then the units digit of n^2+4n+4=(n+2)^2 is 1. If the units digit of a square is 1, then the original number’s units digit is either 1 or 9. But n+2 can’t have a last digit of 9 because then n’s last digit would be 7. So n+2 has a last digit of 1 and so n+3 has a units digit of 2.</p>
<p>^That’s a lot faster than what I did – went through all single-digit odd numbers to test to see if n could equal 7 and only 9 worked. 9+3 = 12, so the units digit of n+3 = 2.</p>