When is a number divisible by another number?
A number is divisible by another number when, after dividing, the remainder is zero.
Examples:
14 is divisible by 7 because 7 goes into 14 two times fully and the remainder is zero.
15 is not divisible by 7 because after dividing, there is a remainder of 1.
remainder
In arithmetic, the remainder is the number "left over" after the division process of two integers.
Example:
When you divide 12 by 5, the remainder is 2.
5 x 2 = 10
12 - 10 = 2
Let's come up with the remainder formula while solving this simple problem.
When an integer is divided by 5, the remainder is 3.
List the smallest five numbers that match the requirement.
An integer, when divided by 5, leaves a remainder of 3. The smallest such number is 8. 8 divided by 5 results in 1 remainder 3. The next number is 13. 13 divided by 5 is 2 remainder 3. The smallest five integers are:
{8, 13, 18, 23, 28}
It looks like those numbers can be found by multiplying 5 by 1, 2, 3, 4 or 5 and adding 3 to the product.
N = D x Q + R
D - divisor; Q - quotient; R - remainder
In the example above, 5 is a divisor, (1,2,3,4,5) are the quotients, and 3 is a remainder.
When is a number divisible by 2?
A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).
Example:
168 is divisible by 2 since the last digit is 8.
When is a number divisible by 3?
A number is divisible by 3 if the sum of its digits is divisible by 3.
Example:
168 is divisible by 3 since the sum of the digits is 15 (1 + 6 + 8 = 15) and 15 is divisible by 3.
When is a number divisible by 4?
A number is divisible by 4 if the number formed by the last 2 digits is divisible by 4.
Example:
316 is divisible by 4 since 16 is divisible by 4.
When is a number divisible by 5?
A number is divisible by 5 if its last digit is 0 or 5.
Example:
185 is divisible by 5.
When is a number divisible by 6?
A number is divisible by 6 if it is divisible by both 2 and 3.
Example:
462 is divisible by 6 because it's divisible by both:
2 (its last digit is even)
and
3 (the sum of its digits is divisible by 3).
When is a number divisible by 7?
*** Note: You don't have to memorize this rule unless you want to surprise your teacher or your peers.
A number is divisible by 7 if you can double its last digit, subtract it from the rest of the number, and the answer is either 0 or divisible by 7.
Example:
812 is divisible by 7.
2 x 2 = 4
81 - 4 = 77
77 ÷ 7 = 11
When is a number divisible by 8?
A number is divisible by 8 if its last three digits form a number that is divisible by 8.
Example:
95,064 is divisible by 8 since the last three digits form a number divisible by 8:
64 ÷ 8 = 8
When is a number divisible by 9?
A number is divisible by 9 if the sum of its digits is divisible by 9.
Example:
3,654 is divisible by 9 since the sum of its digits (3 + 6 + 5 + 4 = 18) is divisible by 9.
When is a number divisible by 10?
A number is divisible by 10 if it ends in 0.
Example:
240 is divisible by 10.
When is a number divisible by 11?
*** Note: You don't have to memorize this rule unless you want to surprise your teacher or your peers. Chances of seeing a "divisibility by 11" problem on the SAT are slim to none.
A number is divisible by 11 if you can add up every second digit, subtract all other digits, and the answer is either 0 or divisible by 11.
Example:
2937 is divisible by 11.
9 + 7 = 16
16 - (2 + 3) = 11
11 is divisible by 11.
When is a number divisible by 12?
A number is divisible by 12 if it is divisible by both 3 and 4.
Example:
516 is divisible by 12 because it's divisible by both:
3 (the sum of its digits is divisible by 3)
and
4 (the number formed by its last two digits, 16, is divisible by 4).
What is the rule that governs the order of operations so that they can be performed correctly?
We perform the operations in the following order:
P – terms in Parentheses
E – Exponents and Roots
M/D – Multiplication and Division
A/S – Addition and Subtraction
The acronym PEMDAS can help students remember the rule.
Which phrase can help you remember the order of operations?
PEMDAS is often expanded to "Please Excuse My Dear Aunt Sally" with the first letter of each word creating the acronym PEMDAS.
Solve this equation using the proper order of operations.
5 x 2 + (24 - 16)^{2} - 18 = ?
5 x 2 + (24 - 16)^{2} - 18 = 56
Parentheses: 24 - 16 = 8
Exponents: 8^{2} = 64
Multiplication: 5 x 2 = 10
Addition/Subtraction: 10 + 64 - 18 = 56
When operations have the same rank according to PEMDAS, as in the example below, what should you do first?
15 - 3 + 28 + 5 - 10 = ?
We know that addition and subtraction operations are treated equally by math in terms of order of operations.
If the operations have the same rank, simply work left to right.
15 - 3 + 28 + 5 - 10 = 35
- 15 - 3 = 12
- 12 + 28 = 40
- 40 + 5 = 45
- 45 - 10 = 35
What are the mean, median, and mode used for?
The mean (or average), median, and mode are used to describe a set of data in which each item is a number.
How do you find the average of a set of numbers?
The arithmetic mean (average) is the sum of the terms divided by the number of the terms.
Average = ^{Sum of Terms}/_{Number of Terms}
Example:
Find the average of the following set of numbers: {3, 17, 4, 16}.
3 + 17 + 4 + 16 = 40
There are 4 items in the set, so
40 ÷ 4 = 10.
10 is the average of the set.
How do you find the sum of terms using the average of terms?
By definition,
Average = ^{Sum of Terms}/_{Number of Terms} ⇒
Sum = (Average) x (Number of Terms)
Example:
The average of 4 numbers is 10. What is the sum of these 4 numbers?
10 x 4 = 40
What numbers are called evenly spaced?
Evenly spaced numbers have the same "space" between them.
For example, consecutive integers are evenly spaced numbers since the difference between any two consecutive terms is 1.
Any arithmetic sequence is a set of evenly spaced numbers.
Suppose you need to find the average of all integers from 5 to 75. There are 71 numbers between them! Adding all of them together would take a long time.
How do you quickly find the average of a series of evenly spaced numbers?
To find the average of a set of evenly spaced numbers, just average the smallest and largest terms.
Example:
Find the average of all integers from 5 to 75.
(5 + 75) ÷ 2 = 40
Logic: there are 71 terms. Adding 1st and 71st, or 2nd and 70th, and so on gives you 35 pairs of 80 and 1 number in the middle that doesn't have a pair. That number is 40.
Suppose there is a set of numbers in which 5 is repeated 3 times, 6 is repeated twice and 4 is repeated 5 times.
How do you find the average of such set of numbers?
Can you add 5, 6 and 4 and divide by 3? Isn't that how you find the average? Yes, it is but NOT the average of a set of numbers where the numbers have different weight, i.e. are repeated more times than others.
Write the set in the expanded form.
{5, 5, 5, 6, 6, 4, 4, 4, 4, 4}
Clearly, you need to find the sum of all terms (5 + 5 +.... 4). Or you write the sum of terms this way:
(5 x 3) + (6 x 2) + (4 x 5)
Now, divide by 10 since there are 10 numbers in the set. The average is 4.7.
FYI, you just found the weighted average of the set above.
Can you formulate how to find the average of a set of numbers with different weights?
Weighted Average = ?
To find the weighted average of a set of numbers:
- Find the product of each term and its weight
- Find the sum of all products
- Find the sum of all terms
- Divide the sum of all products by the sum of all terms
You scored 70 on 3 tests out of 7 this semester and 80 on the rest.
What is your average score on all 7 tests during this semester?
75.71 is the average score on all 7 tests.
Add the scores together to find the total score of all 7 tests. Then, divide the sum by 7 to find the average score.
3 x 70 = 210
4 x 80 = 320
320 + 210 = 530
530 ÷ 7 = 75.71
What is the median of a set of numbers?
The median is the middle value of the set of numbers when placed in order.
How do you find the median of a set with an odd number of elements?
If there is an odd number of elements in a set, the median is the middle number.
Example:
Find the median of this set of numbers: {4, 7, 2, 8, 1, 34, 9}
First, write the numbers in order. Then, find the middle number:
{1, 2, 4, 7, 8, 9, 34}. 7 is the median.
How do you find the median of a set with an even number of elements?
When there is an even number of items, the median is the average of the two middle numbers.
Example:
Find the median of this set of numbers: {34, 38, 52, 14, 20, 36}.
First, reorder the numbers from least to greatest.
{14, 20, 34, 36, 38, 52}
The average of 34 and 36 is 35, which is the median.
What is the mode of a set of numbers?
The mode is the item(s) that occur(s) most frequently.
Example:
Find the mode of the set of numbers below {15, 18, 15, 24, 37, 15, 3, 88, 34}
The mode is 15, as it appears most frequently.