Students taking the SAT must make sure that they are very comfortable with the terms “factor” and “multiple” and, therefore, know the difference between the two. For example, let’s use the number 6. What are the factors of 6 as opposed to the multiples of 6? 

By definition, the factors of 6 are those numbers that divide evenly into 6. Another way of saying this is that 6 is perfectly divisible by these factors. A factor is synonymous with a “divisor.” The factors of 6 are 1, 2, 3 and 6. When asked for the factors of 6, many students would say “1 times 6 and 2 times 3.” We are not being asked to break 6 into factors, but merely to list the factors. The best way to do this is to start with 1 (which divides evenly into all integers) and proceed upward listing all numbers that divide evenly into the given number. That is why we recommend writing the list as 1, 2, 3 and 6.

What are the multiples of 6? Think of the multiplication table:  1 times 6, 2 times 6, 3 times 6, etc. Thus, the multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, … 

The factors of 6 divide evenly into 6.  6 divides evenly into the multiples of 6.

One interesting note: We could have chosen any number for our illustration above but 6 was specifically chosen. Other than the number 6 itself, the other factors of 6 are 1, 2 and 3. Add them up:  1 + 2 +3 = 6, the number itself!!! It is very rare to have a number be the sum of its own factors (other than itself). Such a number is called a “perfect number.” Perfect numbers are few and far between. Challenge: What’s the next perfect number after 6? In other words, it has to be the sum of its own factors (other than itself). Hint: This one is not so big- we’ll tell you it is less than 50.

Here’s a point of algebra that many students have forgotten. What do we do if we are confronted with a division problem involving more than one term (a polynomial) divided by one term (a monomial)? Let’s consider a situation as simple as this: How do we handle the division (a + b)/c? Most students are very comfortable with the concept of the distributive property for multiplication but may not realize that the same thing goes for division. With this problem, we just “distribute” the division.  (a + b)/c is the same as taking each term from the numerator and dividing it by the denominator:  a/c + b/c.  It’s that simple! Now, how would that work on an SAT problem? A recently released exam provides a great example.

In their newest version of The Official SAT Study Guide, Practice Test #2, Section 5, problem #12 on page 467 is a perfect example: “If (3x + y)/y = 6/5, what is x/y?” Now, there are several ways to solve this problem but it screams for a division as described above. The left side of this equation would break into two divisions 3x/y + y/y.  Since y/y = 1, we now have 3x/y + 1 = 6/5. Subtract 1 from both sides to give us 3x/y = 1/5.  Now just divide by 3 and we will have the answer x/y = 1/15. (We’ll just give a hint that another method of solution is to cross-multiply the given expression.)

One of the nicest lessons that a high school math teacher gets to teach concerns the sum of the measures of the interior angles of a polygon. Draw a hexagon (6 sides). Put your pencil on one vertex and draw every possible diagonal originating from that vertex. How many non-overlapping triangles has the hexagon been divided into? If you’ve done it correctly, you’ve got 4 triangles. If you follow this same process with, let’s say, an octagon (8 sides), you’ll find it divided into 6 triangles. Students can easily be led to the logical conclusion that if we have a polygon of “n” sides, then we will be able to divide it into (n – 2) triangles. Thus, if we are trying to discover the sum of the measures of the interior angles of the polygon (knowing that the sum of the measures of the angles of a triangle is 180), we can conclude that our sum must be   (n – 2) times 180. If we are now asked to use this formula to find the sum of the interior angles of a decagon (10 sides), we plug in n=10, giving us (10 – 2), or 8 times 180, for a sum of 1440 for the interior angles. 

That’s a great formula to learn and to know for the SAT. However, the far more important SAT fact regards the exterior angles. What is the formula for the sum of measures of the exterior angles of a polygon? We might jokingly say that it’s not exactly a formula. The sum of the measures of the exterior angles of a polygon is 360. Students always ask how many sides the polygon has. The answer is that the sum of these exterior angles is always 360 regardless of how many sides the polygon has. The sum is not dependent of “n”, the number of sides. The writers of the SAT capitalize on this in a variety of ways. Over the years there have been several problems with a triangle and the exterior angles labeled “x”, “y” and “z”. With absolutely no given information, they ask for the sum of x, y and z. The message is clear. Students taking the SAT must know that the sum of the measures of the exterior angles of ANY polygon is always 360.

Virtually every student knows that the sum of the measures of the three angles of a triangle is 180 (and for those who don’t know this, the kind folks at the Educational Testing Service have listed this fact at the top of every SAT math section).

Consider this question: What is the average measure of one angle of a triangle? It’s a simple question. To find the average of any three things, we find the sum and then divide by 3. Since the sum of the three angles of a triangle is always 180, when we divide by 3 we find that the average measure of an angle of a triangle is always 60. This fairly simple fact has nice implications for SAT math problems.

Only an equilateral triangle has 3 angles that each measure exactly 60.  If we then decrease one of the angles by one degree to 59, to balance this out one of the other angles must increase to 61 so that the average remains 60 (and the sum remains 180). Now we have three unequal angles. Let’s examine them more closely. What can be said about the smallest angle of a triangle? Setting aside the equilateral triangle situation, the smallest angle of a triangle must measure less than 60. Are there any other conclusions that we can draw about the other two angles of the triangle? The largest angle of the triangle must measure greater than 60. Is there anything we can state for sure regarding the “middle-sized” angle of a triangle? It may be more or less than 60, we can’t tell for sure. However, if you give it some further thought, could this angle be as large as 90 degrees? If the “middle-sized” angle of a triangle is 90 and then the largest angle is more than that, we would have more than 180 degrees. Therefore, the “middle-sized” angle must measure less than 90.

All of these conclusions derive directly from the simple fact that the sum of the measures of the angles of a triangle is 180. Make no mistake about it – the SAT will test you about these more in-depth concepts. Taking an SAT Prep Course will hone your math skills.

A guy walks up to you on the street and asks you what happens when you raise a number to a higher power. Most people would naively respond that the number grows larger. However, that’s a very simplistic response that is not always true even though it sounds like a sure-fire result. 

When you take any number greater than one and raise it to the second or third powers or higher, yes indeed, the result is certainly bigger. However, the SAT folks will take advantage of you if you think this holds true for every type of number. 

Consider a positive fraction (or decimal) between zero and one: let’s use ½ or .5. Let’s square it giving us ¼ or .25. The value has actually decreased! Can this happen with any other type of number? Let’s try negative numbers less than -1 such as -2. Let’s raise it to an odd power so that it remains negative. Try -2 to the 3^rd power. This gives us -8 which is smaller than the -2 we started with. 

Another situation to consider is a negative fraction (or decimal) between -1 and zero. Let’s try -1/2 or -.5. We’ll raise it to the 3^rd power so that it again remains negative. -1/2 to the 3^rd is -1/8 or -.125. If you look at a number line, the result is a smaller negative number so that it is really larger than the original value. 

The last situation is to ask if it is possible to raise a number to a higher power and the result is the same as the original number. Yes! Zero to any power is zero. One raised to any power is still one. Also, -1 raised to any odd power will remain -1.

The Educational Testing Service knows how to test your knowledge of various types of numbers raised to various powers.  Familiarize yourself with each of the above situations.