Mathematics: Class 5 — Chapter 13: Animal Jumps

13.1 Introduction to Factors and Multiples

When numbers are multiplied together, they form a product. The numbers being multiplied are called factors, and the result is a multiple of those factors.

Arrays and Factors

An array is a systematic arrangement of objects in rows and columns that helps visualize factors. For example, the number $15$ can be arranged as:

$3 \times 5 = 15$

Here, $3$ and $5$ are the factors, and $15$ is the multiple.

Similarly, we can make different arrays for the number $12$:

$3 \times 4 = 12$
$2 \times 6 = 12$
$1 \times 12 = 12$

Therefore, the factors of $12$ are $1, 2, 3, 4, 6,$ and $12$. Each of these numbers can divide $12$ completely without leaving a remainder.

Key Rule: The number itself is always a multiple of itself.

13.2 Prime Numbers

Some numbers have only two factors: $1$ and the number itself. These are called prime numbers.

Examples: $13$ (factors are $1$ and $13$), $37$ (factors are $1$ and $37$).

13.3 Common Multiples

When two or more animals jump along a number line, the positions they both land on represent their common multiples.

The Rabbit and the Frog

The rabbit takes jumps of $4$ steps each time: landing on $4, 8, 12, 16, 20, 24, \dots$
The frog takes jumps of $3$ steps each time: landing on $3, 6, 9, 12, 15, 18, 21, 24, \dots$
The numbers both of them touch are the common multiples of $3$ and $4$. The first common multiple is $12$. Other common multiples include $24, 36, 48, \dots$

The Spider and the Grasshopper

The spider takes jumps of $3$ steps each time: landing on $3, 6, 9, 12, 15, 18, \dots$
The grasshopper takes jumps of $6$ steps each time: landing on $6, 12, 18, 24, \dots$
The common multiples of $3$ and $6$ are $6, 12, 18, 24, \dots$

13.4 Common Factors

A common factor of two or more numbers is a number that exactly divides each of them without leaving a remainder.

Key Rule: The number itself and 1 are always factors of any number.

Finding Common Factors of 24 and 36

Factors of $24$: $1, 2, 3, 4, 6, 8, 12, 24$
Factors of $36$: $1, 2, 3, 4, 6, 9, 12, 18, 36$
The common factors of $24$ and $36$ are: $1, 2, 3, 4, 6, 12$

13.5 Divisibility and Sorting

Numbers can be sorted into groups based on their divisibility by $2$, $5$, and $10$:

Divisible by 2 only: Even numbers that do not end in $0$ or $5$ (e.g., $22, 38, 66, 78, 62, 84, 56$).
Divisible by 5 only: Numbers ending in $5$ (e.g., $25, 45, 55, 75, 95$).
Divisible by 10 only: This category overlaps with both $2$ and $5$. Any number divisible by $10$ is also divisible by $2$ and $5$.
Divisible by 2, 5, and 10: Numbers ending in $0$ (e.g., $30, 40, 90$).

Mathematics: Class 5 — Chapter 13: Animal Jumps

Arrays and Factors

The Rabbit and the Frog

The Spider and the Grasshopper

Finding Common Factors of 24 and 36

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