Multiples, Common Multiples and Lowest Common Multiples

Resourses:

http://www.mathgoodies.com/Lessons/vol3/lcm.html

http://webmath.com/intlcm.html

http://www.mathsteacher.com.au/year8/ch01_arithmetic/03_mult/mult.htm

Multiples
The multiples of a number are its products with the natural numbers 1, 2, 3, 4, 5, ....
Example 1
1 x 8 = 8
2 x 8 =16
3 x 8 =24
4 x 8 =32
5 x 8 =40
So, the multiples of 8 are 8, 16, 24, 32, 40 and so on.
Note:
The multiples of a number are obtained by multiplying the number by each of the natural numbers.

Example 2
Write down the first five multiples of 9.
Solution:
1 x 9 = 9
2 x 9 =18
3 x 9 =27
4 x 9 =36
5 x 9 =45
The multiples of 9 are obtained by multiplying 9 with the natural numbers 1, 2, 3, 4, 5 …
So, the first five multiples of 9 are 9, 18, 27, 36 and 45.

Common Multiples
Common multiples are multiples that are common to two or more numbers.

Example 3
Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, …
Multiples of 3 are 3, 6, 9, 12, 15, 18, …
So, common multiples of 2 and 3 are 6, 12, 18, …
Example 4
Find the common multiples of 3 and 4.
Solution:
Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, …Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, …
So, the common multiples of 3 and 4 are 12, 24, 36, …

Lowest Common Multiple
The lowest common multiple (LCM) of two or more numbers is the smallest common multiple.
Example 5
Multiples of 8 are 8, 16, 24, 32, …
Multiples of 6 are 6, 12, 18, 24, …
The LCM of 6 and 8 is 24
.
Example 6
Find the lowest common multiple of 2 and 5.
List the multiples of 5 and stop when you find a multiple of 2.
Multiples of 5 are 5, 10, …
Multiples of 2 are 2, 4, 6, 8, 10, …
The LCM of 2 and 5 is 10

NUMBER PATTERNS AND SEQUENCES
Real-Life Examples
Try to discover geometric patterns in nature (shells, flowers, animals, pine cones, rocks); in architecture (archways, doorways, stairways, floor tiles, windows); in clothing and home fashion (designer labels, t-shirt logos, neckties, quilts, wallpaper, and floor tile); and in technology (computer-generated graphics and logos).

Create, Extend and Explain Number Patterns
Have fun with your family by making up number patterns. Encourage other family members to discover each pattern, predict the next three numbers in the pattern, and explain how the pattern works.
5, 10, 15, 20, 25, 30 … (35, 40, 45)
Add 5 to the previous number.
Count by five’s.

2, 4, 6, 8, 10, 12 … (14, 16, 18)
Add 2 to the previous number.
List the even numbers.
Count by 2.

1, 3, 5, 7, 9, 11 … (13, 15, 17)
Add 2 to the previous number.
List the odd numbers.

1, 2, 4, 8, 16 … (32, 64, 128)
Double the previous number.
Multiply the previous number by 2.
Add 1 to the first number, then 2 to the next number, then 4, then 8, etc.

1, 4, 9, 16, 25, 36, 49 … (64, 81, 100)
Multiply each number by itself (1 x 1 = 1, 2 x 2 = 4, 3 x 3 = 9, 4 x 4 = 16).
Find the perfect squares of the counting numbers.
Add 3 to the first number, then 5 to the next number, then 7, then 9, then 11, etc.

Write Words on Calculators?

This is a fun activity to do when you are bored in your Math class!

Calculator Words. Type 200 Words Using A Simple Calculator - These bloopers are hilarious
CALCULATOR WORDS
Use a calculator to find these words by doing the following calculations and then turning your calculator upside down.
a. 3357 -2223 to get a place you wouldn’t go to. (………………..…….)
b. 300 000 + 18 830 to get a girl’s name. (…………………..….)
c. 85 423 + 294 496 to get something you might do if you’re happy or embarrassed about something. (…………………..….)
d. 6411 − 897 to get a noise you wouldn’t want to hear while bushwalking. (………..…………….)
e. 63 552 ÷ 64 to get something you eat. (…………………..….)
f. 203 × 15 to get something you wear. (……………..……….)
g. 52 043 ÷ 71 to get an animal. (………………..…….)
h. 52 360 ÷ 17 to get a musical instrument. (…..………………….)
i. 4417 x 8 to get some animals. (………………..…….)
j. 923 x 5 to get something you need your breath to do. (……..……………….)
k. 23 x 19 + 7301 to get something you often hear at school. (……..……………….)
l. 888 888 ÷ 2 – 65 638 to get something you shouldn’t do when you eat. (……..……..………..)
m. 4536 ÷ 81 + 261 to get something you should never do. (……………….……..)
n. 237 023 x 2 ÷ 421 x 2 + 4853 to get something you find in the garden. (……….……………..)
o. 6716 ÷ 73 x 125 136 ÷ 16 – 642 187 to get something you may find on the beach. (………………….…..)
p. 7 + 700 + 7000 + 10 000 + 300 000 + 5 000 000 to get something that you shouldn’t eat too much of. (……..……………....)

The following videos show more examples of the application of PEMDAS

COMBINED OPERATIONS
If the expression consists of parenthesis, exponents, +, –, × and ÷, then the operations MUST be performed in the following order.
Always work on the calculations within parenthesis first if any.
Next, calculate the exponents.
Then, carry out multiplication or division, working from left to right.
Lastly, do addition or subtraction, working from left to right.

The order to perform combined operations is called the PEMDAS rule.
Note: A common mnemonic for PEMDAS is Please Excuse My Dear Aunt Sally.
Example:
Evaluate 10 ÷ 2 + 12 ÷ 2 × 3
Using the PEMDAS rule, we need to evaluate the division and multiplication before subtraction and addition. It is recommended that you put in parenthesis to remind yourself the order of operation.
Solution:
10 ÷ 2 + 12 ÷ 2 × 3
= ( 10 ÷ 2) + (12 ÷ 2 × 3)
= 5 + 18
= 23

HAPPY NEW YEAR.

WELCOME BACK TO SCHOOL.

........................................................................

WHOLE NUMBERS
Divisibility Tests

Divisibility by 2
A whole number is divisible by 2 if the digit in its units position is even, (either 0, 2, 4, 6, or 8).
Examples:
The number 84 is divisible by 2 since the digit in the units position is 4, which is even.The number 333336 is divisible by 2 since the digit in the units position is 6, which is even.The number 1297000 is divisible by 2 since the digit in the units position is 0, which is even.

Divisibility by 3
A whole number is divisible by 3 if the sum of all its digits is divisible by 3.
Examples:
The number 177 is divisible by three, since the sum of its digits is 15, which is divisible by 3.The number 8882151 is divisible by three, since the sum of its digits is 33, which is divisible by 3.The number 162345 is divisible by three, since the sum of its digits is 21, which is divisible by 3.
If a number is not divisible by 3, the remainder when it is divided by 3 is the same as the remainder when the sum of its digits is divided by 3.
Examples:
The number 3248 is not divisible by 3, since the sum of its digits is 17, which is not divisible by 3. When 3248 is divided by 3, the remainder is 2, since when 17, the sum of its digits, is divided by three, the remainder is 2.
The number 172345 is not divisible by 3, since the sum of its digits is 22, which is not divisible by 3. When 172345 is divided by 3, the remainder is 1, since when 22, the sum of its digits, is divided by three, the remainder is 1.

Divisibility by 4
A whole number is divisible by 4 if the number formed by the last two digits is divisible by 4.
Examples:
The number 3124 is divisible by 4 since the number formed by its last two digits, 24, is divisible by 4.The number 1333336 is divisible by 4 since the number formed by its last two digits, 36, is divisible by 4.The number 1297000 is divisible by 4 since the number formed by its last two digits, 0, is divisible by 4.
If a number is not divisible by 4, the remainder when the number is divided by 4 is the same as the remainder when the last two digits are divided by 4.
Example:
The number 172345 is not divisible by 4, since the number formed by its last two digits, 45, is not divisible by 4. When 172345 is divided by 4, the remainder is 1, since when 45 is divided by 4, the remainder is 1.

Divisibility by 5
A whole number is divisible by 5 if the digit in its units position is 0 or 5.
Examples:
The number 95 is divisible by 5 since the last digit is 5.The number 343370 is divisible by 5 since the last digit is 0. The number 129700195 is divisible by 5 since the last digit is 5.
If a number is not divisible by 5, the remainder when it is divided by 5 is the same as the remainder when the last digit is divided by 5.
Example:
The number 145632 is not divisible by 5, since the last digit is 2. When 145632 is divided by 5, the remainder is 2, since 2 divided by 5 is 0 with a remainder of 2.
The number 7332899 is not divisible by 5, since the last digit is 9. When 7332899 is divided by 5, the remainder is 4, since 9 divided by 5 is 1 with a remainder of 4.

Divisibility by 6
A number is divisible by 6 if it is divisible by 2 and divisible by 3. We can use each of the divisibility tests to check if a number is divisible by 6: its units digit is even and the sum of its digits is divisible by 3.
Examples:
The number 714558 is divisible by 6, since its units digit is even, and the sum of its digits is 30, which is divisible by 3. The number 297663 is not divisible by 6, since its units digit is not even.The number 367942 is not divisible by 6, since it is not divisible by 3. The sum of its digits is 31, which is not divisible by 3, so the number 367942 is not divisible by 3.

Divisibility by 8
A whole number is divisible by 8 if the number formed by the last three digits is divisible by 8.
Examples:
The number 88863024 is divisible by 8 since the number formed by its last three digits, 24, is divisible by 8.The number 17723000 is divisible by 8 since the number formed by its last three digits, 0, is divisible by 8.The number 339122483984 is divisible by 8 since the number formed by its last three digits, 984, is divisible by 8.
If a number is not divisible by 8, the remainder when the number is divided by 8 is the same as the remainder when the last three digits are divided by 8.
Example:
The number 172045 is not divisible by 8, since the number formed by its last three digits, 45, is not divisible by 8. When 172345 is divided by 8, the remainder is 5, since when 45 is divided by 8, the remainder is 5.

Divisibility by 9
A whole number is divisible by 9 if the sum of all its digits is divisible by 9.
Examples:
The number 1737 is divisible by nine, since the sum of its digits is 18, which is divisible by 9.The number 8882451 is divisible by nine, since the sum of its digits is 36, which is divisible by 9.The number 762345 is divisible by nine, since the sum of its digits is 27, which is divisible by 9.
If a number is not divisible by 9, the remainder when it is divided by 9 is the same as the remainder when the sum of its digits is divided by 9.
Examples:
The number 3248 is not divisible by 9, since the sum of its digits is 17, which is not divisible by 9. When 3248 is divided by 9, the remainder is 8, since when 17, the sum of its digits, is divided by 9, the remainder is 8.
The number 172345 is not divisible by 9, since the sum of its digits is 22, which is not divisible by 9. When 172345 is divided by 9, the remainder is 4, since when 22, the sum of its digits, is divided by 9, the remainder is 4.

Divisibility by 10
A whole number is divisible by 10 if the digit in its units position is 0.
Examples:
The number 1229570 is divisible by 10 since the last digit is 0.The number 676767000 is divisible by 10 since the last digit is 0.The number 129700190 is divisible by 10 since the last digit is 0.
If a number is not divisible by 10, the remainder when it is divided by 10 is the same as the units digit.
Examples:
The number 145632 is not divisible by 10, since the last digit is 2. When 145632 is divided by 10, the remainder is 2, since the units digit is 2.The number 7332899 is not divisible by 10, since the last digit is 9. When 7332899 is divided by 10, the remainder is 4, since the units digit is 9.

Divisibility by 11
Starting with the units digit, add every other digit and remember this number. Form a new number by adding the digits that remain. If the difference between these two numbers is divisible by 11, then the original number is divisible by 11.
Examples:
Is the number 824472 divisible by 11? Starting with the units digit, add every other number:2 + 4 + 2 = 8. Then add the remaining numbers: 7 + 4 + 8 = 19. Since the difference between these two sums is 11, which is divisible by 11, 824472 is divisible by 11.
Is the number 49137 divisible by 11? Starting with the units digit, add every other number:7 + 1 + 4 = 12. Then add the remaining numbers: 3 + 9 = 12. Since the difference between these two sums is 0, which is divisible by 11, 49137 is divisible by 11.
Is the number 16370706 divisible by 11? Starting with the units digit, add every other number:6 + 7 + 7 + 6 = 26. Then add the remaining numbers: 0 + 0 + 3 + 1=4. Since the difference between these two sums is 22, which is divisible by 11, 16370706 is divisible by 11.

Divisibility by 12
A number is divisible by 12 if it is divisible by 4 and divisible by 3. We can use each of the divisibility tests to check if a number is divisible by 12: its last two digits are divisible by 4 and the sum of its digits is divisible by 3.
Examples:
The number 724560 is divisible by 12, since the number formed by its last two digits, 60, is divisible by 4, and the sum of its digits is 30, which is divisible by 3.The number 36297414 is not divisible by 12, since the number formed by its last two digits, 14, is not divisible by 4.The number 367744 is not divisible by 12, since it is not divisible by 3. The sum of its digits is 29, which is not divisible by 3, so the number 367942 is not divisible by 3.

Divisibility by 15
A number is divisible by 15 if it is divisible by 3 and divisible by 5. We can use each of the divisibility tests to check if a number is divisible by 15: its units digit is 0 or 5, and the sum of its digits is divisible by 3.
Example:
The number 7145580 is divisible by 15, since its units digit is even, and the sum of its digits is 30, which is divisible by 3.

Divisibility by 16
A whole number is divisible by 16 if the number formed by the last four digits is divisible by 16.
Examples:
The number 898630032 is divisible by 16 since the number formed by its last four digits, 32, is divisible by 16.The number 1772300000 is divisible by 16 since the number formed by its last four digits, 0, is divisible by 16.The number 339122481296 is divisible by 16 since the number formed by its last four digits, 1296, is divisible by 16.
If a number is not divisible by 16, the remainder when the number is divided by 16 is the same as the remainder when the last four digits are divided by 16.
Example:
The number 172411045 is not divisible by 16, since the number formed by its last four digits, 1045, is not divisible by 16. When 172411045 is divided by 16, the remainder is 5, since when 1045 is divided by 16, the remainder is 5.

Divisibility by 18
A number is divisible by 18 if it is divisible by 2 and divisible by 9. We can use each of the divisibility tests to check if a number is divisible by 18: its units digit is even and the sum of its digits is divisible by 9.
Examples:
The number 7145586 is divisible by 18, since its units digit is even, and the sum of its digits is 36, which is divisible by 9. The number 2976633 is not divisible by 18, since its units digit is not even.The number 367942 is not divisible by 18, since it is not divisible by 9. The sum of its digits is 31, which is not divisible by 9, so the number 367942 is not divisible by 9.

Divisibility by 20
A number is divisible by 20 if its units digit is 0, and its tens digit is even. In other words, the last two digits form one of the numbers 0, 20, 40, 60, or 80.
Examples:
The number 3351002760 is divisible by 20, since the number formed by its last two digits is 60.The number 802199730000 is divisible by 20, since the number formed by its last two digits is 0.

Divisibility by 22
A number is divisible by 22 if it is divisible by the numbers 2 and 11. We can use each of the divisibility tests to check if a number is divisible by 22: its units digit is even, and the difference between the sums of every other digit is divisible by 11.
Example:
Is the number 117524 divisible by 22? The units digit is even, so it is divisible by 2. The two sums of every other digit are 4 + 5 + 1 = 10 and 2 + 7 + 1 = 10, which have a difference of 0. Since 0 is divisible by 11, 117524 is divisible by 11. Thus, 117524 is divisible by 22, since it is divisible by both 2 and 11.

Divisibility by 25
A number is divisible by 25 if the number formed by the last two digits is any of 0, 25, 50, or 75 (the number formed by its last two digits is divisible by 25).
Examples:
The number 73224050 is divisible by 25, since its last two digits form the number 50.The number 1008922200 is divisible by 25, since its last two digits form the number 0.