Animal Jumps Chapter Notes | Mathematics (Maths Mela) Class 5 - New NCERT PDF Download

What is a Factor?

• A factor is a number that divides another number exactly, without leaving any remainder.  
• In simple words, if you multiply two numbers to get a product, then both of those numbers are called factors of the product. 

Example:
Factors of 8 are numbers that divide 8 exactly:

• 1 divides 8 exactly (8 ÷ 1 = 8)
• 2 divides 8 exactly (8 ÷ 2 = 4)
• 4 divides 8 exactly (8 ÷ 4 = 2)
• 8 divides 8 exactly (8 ÷ 8 = 1)

So, the factors of 8 are 1, 2, 4, and 8.

How to find factors?

• Start with the number 1 and go up to the number itself.
• Divide the number by each one.
• If the division has no remainder, then that number is a factor.

Did you know?

• Every number has at least two factors—1 and the number itself.  
• All the factors of a number are either less than or equal to the number.
  

Common Factors

When two or more numbers share the same factor, that factor is called a common factor.

Example:
Find common factors of 12 and 18.

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 18: 1, 2, 3, 6, 9, 18

Common factors are the numbers that appear in both lists: 1, 2, 3, and 6.

How to find Common Factors?

1. Find all factors of the first number.
2. Find all factors of the second number.
3. See which factors appear in both lists.

Find the Hidden Numbers

Sometimes, numbers are created by multiplying another number by the same value each time. If we know the final numbers, we can work backwards to find the multiplier. This also teaches us about common factors— numbers that can divide all given numbers exactly.
Imagine a magic box. You put a number inside, the box multiplies it by a fixed number, and the result appears. In this activity, the numbers coming out of the box are 28, 36, 48, and 72.

We ask:
(a) What could the multiplier be?
(b) Could there be more than one multiplier?
(c) What numbers might have been inside the box?

To solve this, we find numbers that divide all four outputs exactly. These are common factors. 

(a)  The multiplier must be a number that divides all 28, 36, 48, and 72.

• 1 divides all of them.
• 2 divides all of them.
• 4 also divides all of them.
  So, the multiplier can be 1, 2, or 4.

(b) Yes. Because all three (1, 2, and 4) work. So there is more than one multiplier.

(c) 

• If multiplier = 1 → numbers inside are 28, 36, 48, 72
• If multiplier = 2 → numbers inside are 14, 18, 24, 36
• If multiplier = 4 → numbers inside are 7, 9, 12, 18

So, the same results can come from different multipliers.

This activity shows that there can be more than one possible multiplier. It also helps us see that factors divide numbers exactly, while multiples are numbers made by multiplying. The same set of numbers can be made in different ways by choosing different multipliers.

How to Identify Factors Using Arrays (Pairs)

• A factor pair of a number is two numbers that multiply together to give that number.

Example, for 12:

• 1 × 12 = 12
• 2 × 6 = 12
• 3 × 4 = 12

So, the factors are 1, 2, 3, 4, 6, and 12.

• Arrays (rows × columns) also show factor pairs. 

Factors using Arrays

One way to find the factors of a number is by arranging objects into arrays - rows and columns. Each arrangement shows a pair of factors that multiply to make the number.

Here, we see the number 15 can be arranged as 3 rows of 5. This tells us 3 × 5 = 15, so 3 and 5 are factors of 15. This can be seen in the following figure:

Similarly, the number 12 can be arranged as 3 × 4, 2 × 6, and 1 × 12. This tells us its factors are 1, 2, 3, 4, 6, and 12. This can be seen in the following figure:

Question for Chapter Notes: Animal Jumps

Try yourself:What are the factors of 24?

• A.

  1, 2, 3, 6, 12, 24
• B.

  1, 2, 3, 4, 6, 8, 12, 24
• C.

  2, 3, 4, 6, 8, 12
• D.

  1, 2, 4, 6, 8, 24

Explanation

Let’s check for 24 step-by-step:

• 1 × 24 = 24 → 1 and 24 are factors.
• 2 × 12 = 24 → 2 and 12 are factors.
• 3 × 8 = 24 → 3 and 8 are factors.
• 4 × 6 = 24 → 4 and 6 are factors.

So, the complete set of factors is:
1, 2, 3, 4, 6, 8, 12, 24

This matches Option A, so A is correct.

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Prime and Composite Numbers 

1. Prime Number 

• A prime number is any natural number that has exactly two factors—1 and the number itself. 
• For example,  13 = 1 × 13. There are no other factors of 13, so 13 is a prime number 

2. Composite Number 

• Every natural number except 1 that has more than two different factors is called a composite number. 
• For example,  check whether 4 is a prime or composite number? 
        4 is a Composite Number because  Factors of 4 are 1, 2, 4 
        It can be divided by 1, 2, and 4. 

Did you Know?

• The number 1 is neither prime nor composite. 
• It is not a prime because it does not have exactly two factors  
• It is not composite because it does not have more than two factors.

Understanding the difference between prime and composite numbers helps us learn more about how numbers work:

What is a Multiple?

• A multiple of a whole number is the product of the number and any counting number.

Example:
Multiples of 3 are:

• 3 × 1 = 3
• 3 × 2 = 6
• 3 × 3 = 9
• 3 × 4 = 12
• and so on...

So, the multiples of 3 are 3, 6, 9, 12, 15, 18, ...

Did You Know 

• Every number is a multiple of 1. 
• The smallest multiple of any number (other than zero) is the number itself. 
• Every multiple of a number is greater than or equal to that number. 
• The number of multiples of a given number is infinite.
  

The smallest number (other than zero) that is a multiple of two or more counting numbers is the least common multiple (LCM) of the numbers. It is the least number which is divisible by all the given numbers.

Animal Jumps

Sol: 

Rabbit jumps in 4’s: 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 48, 52, 60 …
Frog jumps in 3’s: 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 48, 51, 60 …

The numbers that are the same in both are 12, 24, 36, 48, 60…
These are called common multiples of 3 and 4.

We notice that after 12, they keep meeting every 12 steps.
So the first common multiple is 12, and all other common multiples are also multiples of 12.

Sol: 

1. Mowgli jumps by 2 steps: 0, 2, 4, 6, 8, 10, 12, 14 … 50.
So, he meets friends at 4, 12, 14, and 50.
Reasoning: These numbers are all divisible by 2. For example, 12 ÷ 2 = 6 and 50 ÷ 2 = 25. That is why 2 is a factor of these numbers.

2. Mowgli jumps by 3 steps: 0, 3, 6, 9, 12, 15, 18, 21 … 57.
So, he meets friends at 9, 21, 39, and 57.
Reasoning: Each of these numbers can be divided by 3 exactly. For example, 21 ÷ 3 = 7 and 39 ÷ 3 = 13. So 3 is a factor of them.

3. Mowgli jumps by 5 steps: 0, 5, 10, 15, 20, 25, 30, 35 … 60.
So, he meets friends at 25, 30, and 50.
Reasoning: These numbers are multiples of 5. For example, 25 = 5 × 5 and 50 = 5 × 10. That means 5 divides them without remainder.

4. Mowgli jumps by 10 steps: 0, 10, 20, 30, 40, 50, 60 …
So, he meets friends at 30 and 50.
Reasoning: Both 30 and 50 are multiples of 10. For example, 30 = 10 × 3 and 50 = 10 × 5. This shows 10 is a factor of these numbers.

Divisibility Rules

• If a number is a divisor of a second number, then the number is divisible by the first.
  For example, since 3 is a divisor of 6, so 6 is divisible by 3.
• If a number is not prime, then it would be divisible by a number smaller than itself. We can find out by actual division whether a number is divisible by another number or not.
• For example, if we divide 392 by 7, we find that the remainder is 0, so 392 is divisible by 7. In some cases, the process can be shortened by applying certain tests on the digits of the number.

Below, we give some such tests:

1. Divisibility by 2
   A number is divisible by 2 if its units digit is divisible by 2.
   So, a number is divisible by 2 if its units digit is 0, 2, 4, 6 or 8. A number ending in 1, 3, 5, 7 or 9 is not divisible by 2.
   For example, the numbers 120, 3172, 234, 81396, 105098 are all divisible by 2 while numbers like 13, 287, 335, 7091, 28469 are not divisible by 2.
2. Divisibility by 3
   A number is divisible by 3 if the sum of its digits is divisible by 3.
   For example, the number 384 is divisible by 3 and the sum of its digits (3 + 8 + 4) is 15, which is divisible by 3.
   Similarly, the number 217095 is also divisible by 3 as the sum of its digits (2 + 1 + 7 + 0 + 9 + 5) is 24, which is divisible by 3.
   The number 839 is not divisible by 3 as the sum of its digits (8 + 3 + 9) is 20, which is not divisible by 3.
3. Divisibility by 5
   A number is divisible by 5 if its units digit is 0 or 5.
   For example, the numbers like 205, 3075, 2370, 15000 are all divisible by 5. Numbers like 27, 3189, 200053 are not divisible by 5 as they do not end in 0 or 5.
4. Divisibility by 9
   A number is divisible by 9 if the sum of its digits is divisible by 9.
   For example, the number 7326 is divisible by 9, because 7 + 3 + 2 + 6 = 18 and 18 is divisible by 9. The number 27041 is not divisible by 9, because the sum of its digits 2 + 7 + 0 + 4 + 1 = 14, which is not divisible by 9.
5. Divisibility by 10
   A number is divisible by 10 if its units digit is 0.
   For example, the numbers 100, 8390, 9500, 5401,000 are all divisible by 10. Numbers like 209, 703001, 28597, 4385 are not divisible by 10 as their units digit is not 0.
6. Divisibility by 11
   A number is divisible by 11 if the difference of the sum of its digits in odd places and the sum of its digits in even places (starting from the units place) is either 0 or divisible by 11.
   Study the following table.
   
   So, the numbers 3465, 6457, 95986 and 280929 are all divisible by 11.

Below, we give rules to test the divisibility of a number by composite numbers like 4, 6, 8, 12 and 25:

1. Divisibility by 4
   A number is divisible by 4 if the number formed by its digits in tens and units places is divisible by 4.
   For example, in the number 80372, the number 72 formed by the tens and the units digit 2 is divisible by 4. As you can check this number is divisible by 4.
2. Divisibility by 6
   A number is divisible by 6 if it is divisible by both 2 and 3, i.e., it should be an even number and the sum of its digits should be divisible by 3.
   For example, the number 68370 is divisible by 6. It is an even number and also sum of its digits (6 + 8 + 3 + 7 + 0) is 24, which is divisible by 3.
3. Divisibility by 8
   A number is divisible by 8 if the number formed by the digits at the hundreds, tens and units place is divisible by 8.
   Consider the number 207608.
   The number formed by the digits at the hundreds, tens and units place, i.e., 608 is divisible by 8.
   
   ∴ This number is divisible by 8.
   Since 705 is not divisible by 8, so the number 8705 is not divisible by 8.
   
4. Divisibility by 25
   A number is divisible by 25 if the number formed by the digits at the tens and units places is divisible by 25.
   For example, in the numbers 8750, 23275, 8926825, we observe that 50, 75 and 25, i.e., the numbers formed by the digits at the tens and units places, are divisible by 25. So, all these numbers are divisible by 25.
   

From the above rules, we observe that:
A number is divisible by another number if it is divisible by its co-prime factors.
Thus, number divisible by 2 and 5 will also be divisible by 10.

Sorting the Numbers

Sort the following numbers into those that are—

(a) divisible by 2 only
(b) divisible by 5 only
(c) divisible by 10 only
(d) divisible by 2, 5, and 10.

Sol: ​​​​

(a) Divisible by 2 only
​These numbers are even but not ending in 0 (so not divisible by 5 or 10).
22, 38, 66, 78, 62, 84, 56
​
​(b) Divisible by 5 only
​These numbers end in 5 but are not divisible by 2 (so not divisible by 10 either).
45, 75, 25, 95, 55
​
​(c) Divisible by 10 only
​These numbers end in 0, so are divisible by 10, but not divisible by both 2 and 5 together.
If a number ends in 0, it is divisible by 2 and 5 too. So there is no number divisible only by 10.

(d) Divisible by 2, 5, and 10
​Numbers ending in 0 → divisible by 2, 5, and 10.
90, 30, 40

​​​​