Divisibility, Mean, Median, and Mode Flashcards

In this deck you will find basic formulas for addition, subtraction, multiplication, and division as well as the divisibility rules up to 12. You will review the concepts of mean, median, and mode using examples and practice questions.

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1
Q

When is a number divisible by another number?

A

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.

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2
Q

Define:

remainder

A

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

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3
Q

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.

A

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.

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4
Q

When is a number divisible by 2?

A

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.

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5
Q

When is a number divisible by 3?

A

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.

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6
Q

When is a number divisible by 4?

A

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.

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7
Q

When is a number divisible by 5?

A

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

Example:

185 is divisible by 5.

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8
Q

When is a number divisible by 6?

A

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).

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9
Q

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

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

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10
Q

When is a number divisible by 8?

A

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

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11
Q

When is a number divisible by 9?

A

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.

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12
Q

When is a number divisible by 10?

A

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

Example:

240 is divisible by 10.

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13
Q

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

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.

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14
Q

When is a number divisible by 12?

A

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).

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15
Q

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

A

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.

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16
Q

Which phrase can help you remember the order of operations?

A

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

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17
Q

Solve this equation using the proper order of operations.

5 x 2 + (24 - 16)2 - 18 = ?

A

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

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18
Q

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

15 - 3 + 28 + 5 - 10 = ?

A

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
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19
Q

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

A

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

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20
Q

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

A

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.

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21
Q

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

A

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

22
Q

What numbers are called evenly spaced?

A

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.

23
Q

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?

A

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.

24
Q

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?

A

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.

25
Q

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

Weighted Average = ?

A

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
26
Q

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?

A

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

27
Q

What is the median of a set of numbers?

A

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

28
Q

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

A

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.

29
Q

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

A

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.

30
Q

What is the mode of a set of numbers?

A

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.

31
Q

Is 15,640 divisible by 8?

A

15,640 is divisible by 8 because

640 ÷ 8 = 80.

*** Reminder: a number is divisible by 8 if its last three digits form a number divisible by 8.

32
Q

A number that is divisible by 12 must be divisible by:

(a) 4
(b) 3
(c) 6
(d) all of the above

A

(d) all of the above

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.

33
Q

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

A

(a) 9

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.

34
Q

Find the median of the following set of numbers:

{29, 9, 9, 5, 99, 9, 19, 25, 5, 0, 0}

A

The median is 9.

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.

35
Q

Find the mode of the following set of numbers:

{29, 9, 9, 5, 99, 9, 19, 25, 5, 0, 0}

A

The mode is 9.

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.

36
Q

Find the average of the following set of numbers:

{29, 7, 11, 6, 2, 14, 11, 25, 5, 0}

*** Don’t use the calculator.

A

The average of the set is 11.

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

37
Q

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?

A

18 songs

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

38
Q

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?

A

You need to get 2,450 on your next attempt.

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

39
Q

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?

A

The average of the new set of numbers is w + 5.

  • 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 (wy ) is 6. It also equals w + 5.

40
Q

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?

A

The value of the second number is16.

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

41
Q

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?

A

The average purchase price is $16.

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

42
Q

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?

A

The mean and the median of the set of all possible values of X is 3.5.

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

43
Q

Addend + Addend = Sum

How do you find a missing addend?

If a + b = c, how do you find a or b?

A

a = c - b

b = c - a

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.

44
Q

Minuend - Subtrahend = Difference

If a - b = c,

  • how do you find a?
  • how do you find b?
A

a = b + c

b = a - c

  • Example:*
  • a* - 75 = 15
  • a* = 75 + 15 = 90

90 - b = 15

b = 90 - 15 = 75

45
Q

Multiplicand x Multiplier = Product

If a * b = c,

how do you find a or b?

A

a = c ÷ b

b = c ÷ a

Multiplicand and Multiplier are also known as factors.

  • Example:*
  • a* * 12 = 36
  • a* = 36 ÷ 12 = 3

3 * b = 36

b = 36 ÷ 3 = 12

46
Q

Dividend ÷ Divisor = Quotient

If a ÷ b = c,

  • how do you find a?
  • how do you find b?
A

a = c * b

b = a ÷ c

  • Example:*
  • a* ÷ 12 = 5
  • a* = 12 * 5 = 60

60 ÷ b = 5

b = 60 ÷ 5 = 12

47
Q

What rule should you follow when you add fractions?

a/b + c/d = ?

A

Find common denominator as shown in the formula below:

48
Q

What rule should you follow when you subtract fractions?

a/b - c/d = ?

A

Find the common denominator as shown in the formula below:

49
Q

What rule do you follow when you multiply fractions?

a/b x c/d = ?

A

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

50
Q

What rule should you follow when you divide fractions?

a/b ÷ c/d = ?

A

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: