Divisibility, Mean, Median, and Mode

Question 1

When is a number divisible by another number?

Answer 1

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.

Question 2

Define:

remainder

Answer 2

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

Question 3

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.

Answer 3

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.

Question 4

When is a number divisible by 2?

Answer 4

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.

Question 5

When is a number divisible by 3?

Answer 5

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.

Question 6

When is a number divisible by 4?

Answer 6

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.

Question 7

When is a number divisible by 5?

Answer 7

A number is divisible by 5 if its last digit is 0 or 5.

Example:

185 is divisible by 5.

Question 8

When is a number divisible by 6?

Answer 8

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).

Question 9

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.

Answer 9

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

Question 10

When is a number divisible by 8?

Answer 10

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

Question 11

When is a number divisible by 9?

Answer 11

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.

Question 12

When is a number divisible by 10?

Answer 12

A number is divisible by 10 if it ends in 0.

Example:

240 is divisible by 10.

Question 13

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.

Answer 13

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.

Question 14

When is a number divisible by 12?

Answer 14

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).

Question 15

What is the rule that governs the order of operations so that they can be performed correctly?

Answer 15

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.

Question 16

Which phrase can help you remember the order of operations?

Answer 16

PEMDAS is often expanded to “Please Excuse My Dear Aunt Sally” with the first letter of each word creating the acronym PEMDAS.

Question 17

Solve this equation using the proper order of operations.

5 x 2 + (24 - 16)² - 18 = ?

Answer 17

5 x 2 + (24 - 16)² - 18 = 56

Parentheses: 24 - 16 = 8

Exponents: 8² = 64

Multiplication: 5 x 2 = 10

Addition/Subtraction: 10 + 64 - 18 = 56

Question 18

When operations have the same rank according to PEMDAS, as in the example below, what should you do first?

15 - 3 + 28 + 5 - 10 = ?

Answer 18

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

1. 15 - 3 = 12
2. 12 + 28 = 40
3. 40 + 5 = 45
4. 45 - 10 = 35

Question 19

What are the mean, median, and mode used for?

Answer 19

The mean (or average), median, and mode are used to describe a set of data in which each item is a number.

Question 20

How do you find the average of a set of numbers?

Answer 20

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.

Question 21

How do you find the sum of terms using the average of terms?

Answer 21

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

Question 22

What numbers are called evenly spaced?

Answer 22

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.

Question 23

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?

Answer 23

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.

Question 24

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?

Answer 24

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.

Question 25

Can you formulate how to find the average of a set of numbers with different weights? **Weighted Average** = ?

Answer 25

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

Question 26

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?

Answer 26

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

Question 27

What is the **median** of a set of numbers?

Answer 27

The **median** is the middle value of the set of numbers when placed in order.

Question 28

How do you find the median of a set with an *odd* number of elements?

Answer 28

If there is an *odd* number of elements in a set, the median is the *middle* number. ## Footnote *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.

Question 29

How do you find the median of a set with an *even* number of elements?

Answer 29

When there is an *even* number of items, the median is the *average of the two middle numbers*. ## Footnote *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.

Question 30

What is the **mode** of a set of numbers?

Answer 30

The **mode** is the item(s) that occur(s) most frequently. ## Footnote *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.

Question 31

Is 15,640 divisible by 8?

Answer 31

15,640 is divisible by 8 because 640 ÷ 8 = 80. ## Footnote \*\*\* Reminder: a number is divisible by 8 if its last three digits form a number divisible by 8.

Question 32

A number that is divisible by 12 must be divisible by: (a) 4 (b) 3 (c) 6 (d) all of the above

Answer 32

(d) all of the above ## Footnote To be divisible by 12, a number must be divisible by both 4 and 3. A number that is divisible by 12 is by definition also divisible by 6.

Question 33

The sum of the digits of the product of 30 x 60 x 20 x 40 x 50 is divisible by (a) 9 (b) 4 (c) 6 (d) 8 (e) all of the above

Answer 33

(a) 9 ## Footnote If your answer is (e), re-read the question! You are *not* asked to evaluate the product, but rather the *sum of the digits* of the product. Since 60 = 30 x 2, the expression has two factors of 30. This means that the product must be divisible by 9. Since the product is divisible by 9, it means that the *sum of its digits* must be divisible by 9 - the rule in reverse.

Question 34

Find the median of the following set of numbers: {29, 9, 9, 5, 99, 9, 19, 25, 5, 0, 0}

Answer 34

The median is 9. ## Footnote The median is the middle value of the set. To find it, rewrite the numbers from least to greatest: {0, 0, 5, 5, 9, *9*, 9, 19, 25, 29, 99} This set has an odd number of items. The median is the exact middle value, 9.

Question 35

Find the mode of the following set of numbers: {29, 9, 9, 5, 99, 9, 19, 25, 5, 0, 0}

Answer 35

**The mode is 9.** ## Footnote The mode is the number that occurs most frequently. {0, 0, 5, 5, 9, 9, 9, 19, 25, 29, 99} 5 appears twice, while 9 appears three times.

Question 36

Find the average of the following set of numbers: {29, 7, 11, 6, 2, 14, 11, 25, 5, 0} ## Footnote \*\*\* Don't use the calculator.

Answer 36

**The average of the set is 11.** ## Footnote To find the mean/average of a set of numbers, add all elements in the set and divide by the number of elements. Mentally combine complimentary numbers for easy calculations. 29 + 11 + 7 + 2 + 11 + 6 + 14 + 25 + 5 = 110 There are 10 items in the set. 110 ÷ 10 = 11

Question 37

Your favorite singer has recorded 60 songs on 5 albums. With the release of her latest album, the average number of songs per album has increased to 13. How many songs are on her lastest album?

Answer 37

18 songs ## Footnote The singer has recorded a total of 6 albums including the latest one. The average number of songs per album is 13. The total number of songs is 6 x 13 = 78 Subtract from this total the number of songs on her first 5 albums: 78 - 60 = 18

Question 38

Suppose you took the SAT twice and received the following overall scores: 2,100 and 2,200. If you want your average to be 2,250, what is the score you need to get on your third attempt?

Answer 38

You need to get 2,450 on your next attempt. ## Footnote You want to achieve 2,250 average score on three tries. By definition of the average, the sum of all your scores must be 2,250 x 3 = 6,750. 6750 - 2,100 - 2,200 = 2,450

Question 39

The average of a set of *x* numbers is *w*. If each number is increased by 5, what is the average of the new set of numbers?

Answer 39

The average of the new set of numbers is *w + 5*. ## Footnote *Note*: If every number in a set is increased by a certain value, the average of the set will also increase by that value. *Example:* Set *x*: {0, 1, 2} The average *(w)* of the set above is 1. Set *y*: {5, 6, 7} The average of this set *(w_y *) is 6. It also equals *w* + 5.

Question 40

The average value of three numbers is 9. If twice the sum of the first and the third number equals 22, what is the value of the second number?

Answer 40

The value of the second number is16. ## Footnote Let's call the numbers *A, B* and *C*. Since 9 is the average of three numbers, by definition the sum of *A + B + C* = 27. 2 (*A + C*) = 22 ⇒ *A + C* = 11. 27 - 11 = 16

Question 41

You bought three t-shirts at $8 each, two pairs of jeans at $25 each, and a pair of shoes for $22. What is the average purchase price you paid?

Answer 41

The average purchase price is $16. ## Footnote Each item is bought in different quantities; i.e. has a different weight. Remember how to find the weighted average of a set of numbers? First, find the sum of the products of the price and the quantity of each item. (3 x 8) + (2 x 25) + (1 x 22) = 96 Divide that sum by the total number of items you bought. 96 ÷ 6 = 16

Question 42

*X* and *Y* are positive integers where *X* \< *Y*. The arithmetic mean of *X* and *Y* is 7. What is the mean and the median of the set of all possible values of *X*?

Answer 42

The mean and the median of the set of all possible values of *X* is 3.5. ## Footnote Since the mean of *X* and *Y* is 7, the sum of *X* and *Y* is 14. Since X \< Y, all possible values of X are as follows: {1, 2, 3, 4, 5, 6} Mean: (1 + 2 + 3 + 4 + 5 + 6) ÷ 6 = 21 ÷ 6 = 3.5 Median: (3 + 4) ÷ 2 = 3.5

Question 43

Addend + Addend = Sum How do you find a missing addend? If **a + b = c**, how do you find **a** or **b**?

Answer 43

**a = c - b** *b = c - a* ## Footnote *Example*: a + 15 = 75 a = 75 - 15 You can always think of variables as numerals. Instead of *a + b = c*, think 3 + 5 = 8. If one of the addends is missing, you must subtract the other from the sum.

Question 44

Minuend - Subtrahend = Difference If **a - b = c**, * how do you find **a**? * how do you find **b**?

Answer 44

**a = b + c** *b = a - c* ## Footnote *Example:* *a* - 75 = 15 *a* = 75 + 15 = 90 90 - *b* = 15 *b* = 90 - 15 = 75

Question 45

Multiplicand x Multiplier = Product If **a \* b = c**, how do you find **a** or **b**?

Answer 45

**a = c ÷ b** **b = c ÷ a** ## Footnote Multiplicand and Multiplier are also known as factors. Example: a * 12 = 36 a = 36 ÷ 12 = 3 3 * b = 36 b = 36 ÷ 3 = 1

Question 46

Dividend ÷ Divisor = Quotient If **a ÷ b = c**, * how do you find **a**? * how do you find **b**?

Answer 46

**a = c \* b** **b = a ÷ c** ## Footnote *Example:* *a* ÷ 12 = 5 *a* = 12 \* 5 = 60 60 ÷ *b* = 5 *b* = 60 ÷ 5 = 12

Question 47

What rule should you follow when you add fractions? **^a/_b + ^c/_d** = ?

Answer 47

Find common denominator as shown in the formula below:

Question 48

What rule should you follow when you subtract fractions? **^a/_b - ^c/_d** = ?

Answer 48

Find the common denominator as shown in the formula below:

Question 49

What rule do you follow when you multiply fractions? **^a/_b x ^c/_d** = ?

Answer 49

Multiply the numerators, then multiply the denominators as shown in the formula below:

Question 50

What rule should you follow when you divide fractions? **^a/_b ÷ ^c/_d** = ?

Answer 50

To divide one fractions by another fraction, you need to multiply the first fraction by the reciprocal of the second fraction as shown in the formula: