Page 145 - Math_Genius_V1.0_C6_Flipbook

E:\Working\Focus_Learning\Math_Genius-6_(07-11-2024)\Open_Files\07_Chapter_5\Chapter_5
\ 11-Nov-2024 Bharat Arora Proof-8 Reader’s Sign _______________________ Date __________

(b) If the jump size is 5 or 7, will the frog be able to catch the grasshopper?

Look at the multiplication tables of 5 and 7 and check whether 24 comes in these tables.
5 × 4 = 20 5 × 5 = 25

7 × 3 = 21 7 × 4 = 28

In both the tables, 24 doesn’t come so the frog will not be able to catch the grasshopper.
Hence, the numbers 1, 2, 3, 4, 6, 8, 12 and 24 are called the factors or divisors of 24.

Divisors and Proper Divisors
Divisors of a number are all possible factors of the given number. The divisors of 6 are 1, 2, 3 and 6.
In proper divisors, the number itself is not counted. Proper divisors of 6 are 1, 2 and 3.
Suppose there are two grasshoppers sitting at points 24 and 36, and the frog wants to eat both
of them and takes a jump of 8. Will the jump size of 8 land him on both the positions?

8 16 24 32 40

Obviously, the frog can land on 24 but not on 36. So, what jump size should the frog try?

Let’s once again take the jump size of 4 and 6 to check whether the frog can catch the
grasshopper or not.

4 8 12 16 20 24 28 32 36

6 12 18 24 30 36.

Here, the frog will be able to catch both grasshoppers.
The jump size that will help the frog catch the grasshoppers is the common factor of the
numbers. Hence, 4 and 6 are the common factors of 24 and 36.

Example 5: Find the common factors of 18, 42 and 60.

Solution: Factors of 18 are 1, 2, 3, 6, 9 and 18. Quick Check
Factors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. How many common factors do
the numbers 63 and 112 have?
Factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.

Common factors are 1, 2, 3 and 6.

Even and Odd Numbers

The numbers that are exactly divisible by 2 are called even numbers. All multiples of 2 are called
even numbers. For example, 12, 34, 106, 1358, 4710, etc. are even numbers.

The numbers that are not exactly divisible by 2 are called odd numbers. For example, 21, 45, 619,
207, etc. are odd numbers.

There are infinite even/odd numbers.
An even number always has the digit 0, 2, 4, 6 or 8 in its ones place and an odd number always has
the digit 1, 3, 5, 7 or 9 in its ones place.

143 Prime Time