Bubble gums are sold in packets. A packet of bubble gum contains 9 gums.
If you want 27 gums, you would simply buy three such packets because 27 = 9 x 3. But what if you want to buy 22 gums?
As gums are sold in packets, either I can buy 18 = 9 x 2 or 27 = 9 x 3 bubbles. Hence, 22 is not a multiple of 9.
An integer is the multiple of another integer if it is perfectly divided by it
Integers are special numbers that include both positive and negative numbers. Positive numbers are like happy friends, showing how many of something you have or counting things you enjoy. For example, if you have 3 cookies, you can use a positive number to represent that. Negative numbers, on the other hand, are like the opposite of positive numbers. They represent things like debts or things you owe. For example, if you owe someone 2 dollars, you can use a negative number to show that. Integers also include the number zero, which is like a neutral friend. It represents having neither more nor less of something. Integers are helpful when we want to show gains or losses, compare numbers, or solve math problems. They are like a team of numbers with positive, negative, and neutral members, working together to help us understand different situations in our everyday lives.
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Suppose now you are the manufacturer of bubble gums. You want to find how you can equally divide 27 bubble gums.
Either you can make 27 packets of 1 bubble each, 9 packets of 3 bubbles each, 3 packets of 9 bubbles each or 1 packet of 27 bubbles.
But notice that you cannot equally divide 27 bubbles into 4 packets as 27÷4 = 6.75. There is another way to see this: skip counting.
You find the multiples of 4: 4, 8, 12, 16, 20, 24, 28 … and realize that 27 is skipped so it cannot be a multiple of 4.
An integer is a factor of another integer if it perfectly divides it.
Imagine you have a treasure chest full of shiny jewels. Each jewel represents a special number. Now, imagine that some jewels have secret powers – they are called integers. These special jewels can do something extraordinary – they can divide other jewels perfectly. It’s like having a magical key that unlocks a secret door. When an integer jewel divides another jewel, it means it can be shared without any leftovers or remainders. For example, if the number jewel in the chest is 12, its integer jewels are 1, 2, 3, 4, 6, and 12. These integer jewels can divide 12 perfectly because they can be arranged to form equal groups. Let’s say you have 12 candies, and you want to share them with your friends. You can use the integer jewels to divide the candies equally, so each friend gets the same number of candies without any leftovers.
The integer jewels are like the superheroes of division, making things fair and equal. They help us solve puzzles, share things, and unlock the mysteries hidden within numbers. So, remember, when an integer jewel perfectly divides another jewel, it means they are magical friends that work together to bring harmony and fairness to our number world.
What are the multiples and factors of 15?
Multiples: 15 , 30, 45, 60, 75…
Factors: 1, 3, 5, 15
Do you see any relationship between multiples and factors?
If the first number is the factor of the second, the second is the multiple of the first, and vice versa.
When we say that one number is a factor of another number, it means that it can divide the other number evenly without any leftover parts. It’s like when you have a pizza and you can share it equally among your friends without any leftover slices. On the other hand, when we say that one number is a multiple of another number, it means that you can keep adding the first number to itself to get the second number. For example, if you have 3 candies and you keep adding 3 more candies, you will have 6 candies, 9 candies, and so on.
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So, if the first number is a factor of the second number, it means the second number is a multiple of the first number. And if the second number is a multiple of the first number, it means the first number is a factor of the second number. It’s like having a special relationship between the numbers, where they can be connected through division or multiplication.
We see that 15 is a multiple of 5 and 5 is a factor of 15.
1. Write down the first five multiples of the following numbers:
a) 7 b) 13
c) 22 d) 85
2. Identify which of the following numbers are factors of 20.
1 2 3 4 5 6 8 10
Note: Marina and Shoaib jog everyday in the nearby park. It takes 6 and 9 minutes for Marina and Shoaib to complete one lap respectively. When will they meet at the starting point?
Notice that after 18 minutes, Marina completes her third lap while Shoaib completes his second lap. Hence, they meet right after 18 minutes at the starting point.
18 would then be called the lowest common multiple of 6 and 9.
The lowest common multiple (LCM) of two numbers is the smallest number that is the multiple of both the numbers.
Uswa wants to determine the ratios of the ages of her niece and nephew using numbers as small as possible. Her niece is 15 while her nephew is 10 years old respectively. How would she do that? She lists down the factors of 10 and 15 to find the common factors.
10: 1, 2, 5, 10
15: 1, 3, 5, 15
She sees that 5 is the highest number that divides both 10 and 15.
Highest Common Factor (HCF) of two numbers is the highest number that divides both of them. Hence 5 is the HCF of 15 and 10.
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The Highest Common Factor, or HCF, is like a superhero that helps us find the biggest common factor between two or more numbers. Think of it as a detective that searches for the largest number that can divide evenly into all the given numbers. For example, let’s say we have two numbers, 12 and 18. The HCF is the largest number that can divide both 12 and 18 without leaving any leftovers. In this case, the HCF is 6 because it is the biggest number that can go into both 12 and 18, making both of them divisible without any remainders. The HCF helps us find common factors and simplifies things. It’s like finding the greatest common friend who can help solve problems by identifying the biggest number that can divide multiple numbers evenly. The HCF is a special tool that allows us to work with numbers in a simpler way, just like a helpful superhero with a keen eye for the biggest shared factor!
1. Find the LCM of the following pairs of numbers.
2. Find the HCF of the following numbers.
Note: We have learnt one method to find the LCM and the HCF. Now we will use another method to find the LCM and HCF called the ladder or grid method. Suppose we have to find the LCM of 12 and 30.
In order to find the HCF of 12 and 30 we repeat steps 1 to 4.
Note: In this lesson we will learn about the divisibility rules of 2, 3, 4, 5, 6, 8, 9 and 10.
A number is divisible by 2 only if its last digit is an even number. A number is divisible by 4 only if its last two digits are divisible by 4.
For example: 64 is divisible by 2 because 4 is an even number. it is also divisible by 4 because 64 is divisible by 4.
If a number’s last digit is an even number like 0, 2, 4, 6, or 8, then that number is divisible by 2. It’s like having a special pattern where every other number can be evenly divided by 2. Now let’s move on to the number 4. To check if a number is divisible by 4, we need to look at its last two digits. If those two digits together form a number that is divisible by 4, then the original number is also divisible by 4. For example, if the last two digits are 12 or 24 or 36, then the whole number is divisible by 4. It’s like finding a hidden code where the last two digits must follow a certain rule for the number to be divisible by 4. These rules make it easier for us to quickly figure out if a number can be divided evenly by 2 or 4.
A number will be divisible by 8 if the last 3 digits of the number are divisible by 8.
Divisibility rules help us find out if a number can be divided by another number without any remainders. Let’s learn about the number 8. If we want to know if a number is divisible by 8, we just need to look at its last three digits. If those three digits together make a number that is divisible by 8, then the whole number is divisible by 8 too! For example, if the last three digits are 120, 352, or 888, then the whole number is divisible by 8. It’s like having a special trick where we check if the last three digits pass the test for being divisible by 8. This rule makes it easier for us to quickly figure out if a number is divisible by 8 or not.
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For example: in 5 128 the last three digits 128 are divisible by 8 because 8 x 16 = 128.
Any number is divisible by 5 if it ends with a 0 or 5 and any number is divisible by 10 if it ends with a 0.
For the number 5, all we need to do is look at the last digit. If the last digit is either 0 or 5, then the number can be divided by 5 without any remainders! For example, numbers like 10, 15, or 20 have a last digit of 0 or 5, so they can be divided by 5. Now, for the number 10, it’s even simpler! If a number ends with a 0, then it can be divided by 10 without any remainders. Just like that, numbers like 20, 50, or 100 can be divided by 10. These rules help us quickly figure out if a number can be divided by 5 or 10. It’s like finding a special pattern at the end of the number that tells us if it can be evenly divided.
1. Find out whether each of the following numbers are divisible by 2, 3, 4, 5, 6, 8, 9 and 10. The first one has been done for you.
| Numbers | 2 | 3 | 4 | 5 | 6 | 8 | 9 | 10 |
| 9 |  | | ||||||
| 63 | ||||||||
| 28 | ||||||||
| 195 | ||||||||
| 255 | ||||||||
| 227 | ||||||||
| 1 125 | ||||||||
| 3 000 |
