An integer is a number that is neither a fraction nor a decimal. Think of integers as counting numbers that you see on a number line. Zero is also an integer. <?xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" />
Positive numbers are greater than zero, and negative numbers are less than zero. Zero is neither positive nor negative; it's usually referred to as neutral. Adding a negative number is the same as subtracting, and subtracting a negative number is the same as adding.
Even numbers are divisible by 2, and odd numbers are not. (Zero is an even number.) Only integers can be even or odd.
If you ask the average person how many digits there are in the world, he'll usually say something like "an infinite number." In fact, there are only 10 digits (0 through 9), from which we make an infinite number of numbers. For example, 305 is a three-digit number. 3 is the hundreds digit, 0 is the tens digit, and 5 is the ones or units digit.
If one number divides evenly into another, then the first number is a factor of the second. For example, 3 is a factor of 12. Conversely, 12 is a multiple of 3. To find the factors of a number, list them in pairs. The pairs of numbers whose product is 12 are: 1 x 12, 2 x 6, 3 x 4. Therefore, the factors of 12 are: {1,2,3,4,6, and 12}. The multiples of 12 are: {12, 24, 36, 48, 60...}.
If number x is divisible by number y , number y
goes into number x evenly. Therefore, all multiples are
divisible by their factors. For example, 12 is divisible by
6.
There are several divisibility rules that may prove useful to you
on the exam because you can't use a calculator. We will cover
them in the next screen.
If a number is not divisible by another, there will be a remainder left over. If you divide 26 by 7, there is a remainder of 5.
Prime numbers are divisible by only themselves and 1. By definition, 1 is NOT prime. Also, only positive numbers are prime. The smallest prime number (and the only even prime number) is 2. Rememember factor trees? Use these to split any number into its prime factors. Every number "branches" into two factors; every branch ends in a prime number. Let's use 72 as an example:
Consecutive
simply means in order by either increasing or
decreasing size.
A set of consecutive integers might look like this:
{-3,-2,-1,0,1,2...}
A set of consecutive even integers might look like this:
{-4,-2,0,2,4,6...}
A set of consecutive multiples of 3 might look like this:
{-9,-6,-3,0,3,6...}
As you may have ascertained, distinct is just a mathematical term for 'different'. If a question stipulates that a and b are distinct integers, they can't have the same value. For example, the distinct prime factors of 72 are 2 and 3 because you only want to count one 2 and one 3.
The absolute value of a number is how far away this number is from zero on a number line. The absolute value is always positive.
2
An integer is divisible by two whenever its last digit is divisible by 2. For example 234,926,524 is divisible by 2 because 4 is.
3
An integer is divisible by 3 whenever the sum of its digits is also a multiple of 3. For instance 523,874,944,125 is divisible by 3 because 5+2+3+8+7+4+9+4+4+1+2+5 = 54. If you're not sure whether the number you wind up with is divisible by 3, don't worry; just use the rule again! (5 + 4 = 9, which is divisible by 3, so 54 is, which means the big number must be, too.)
4
An integer is divisble by 4 whenever its last two digits, taken as a two-digit number, is divisible by 4. For example, 234,926,524 is divisible by 4 because 24 is.
5
An integer is divisible by 5 whenever its last digit is either a 5 or a 0.
6
An integer is divisible by 6 whenever it's divisible by both 2 and 3, since 6's prime factorization is 2 * 3. For instance 123,678 is divisible by 6 because its last digit is 8 (so it's divisible by 2) and the sum of its digits is 27 (so it's divisible by 3).
9
An integer is divisible by 9 whenever the sum of its digits is also a multiple of 9. For instance 523,874,944,125 is divisible by 9 because 5+2+3+8+7+4+9+4+4+1+2+5 = 54. If you're not sure whether the number you wind up with is divisible by 9, don't worry; just use the rule again! (5 + 4 = 9, which is divisible by 9, so 54 is, which means the big number must be, too.)
10
An integer is divisible by 10 whenever it ends in a 0. If you think about it, you'll see that this is really just a short way of saying that a number is divisible by both 5 (ends in a 5 or a 0) and 2 (ends in an even number).
Other numbers
Like 6 and 10, other numbers have rules that really are just checking to see whether a number is divisible by two other numbers. Numbers that are divisible by 15, for instance, are also divisible by 3 and 5. But as much fun as it may seem to dream up new rules of your own, you should resist the temptation. When you have to reduce fractions involving large numbers, it pays to just use the little rules (2,3,5,9,10) repeatedly, rather than trying to reduce all in one giant step.
If a sequence of 8 consecutive even integers with increasing values has 36 as its eighth term, then the least number in the sequence is
50
48
24
22
20
How many prime numbers are integer multiples of 16?
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Remembering these basic rules can make your life much easier:
- Even x Even = Even
- Even x Odd = Even
- Odd x Odd = Odd
- Even + Even = Even
- Even + Odd = Odd
- Odd + Odd = Even
- Positive x Positive = Positive
- Positive x Negative = Negative
- Negative x Negative = Positive
If m and n are distinct prime numbers, which of the following can be the sum of m and n?
That's correct. The only even prime number is 2. Since the sum of two odd numbers is always even, you are most likely looking for an even number. 4 and 12 are the only even answer choices. 4 is 2 + 2, but those aren't distinct. So you're left with 12. Since 12 = 5 + 7, you've found the answer.
If x is an even multiple of 3 and y is a multiple of 5, which of the following could be the value of xy?
Good! If x is an even multiple of 3 and y is a multiple of 5, then the smallest xy could be is (2)(3)(5) = 30.
