Exercise 6.3.5
60 | |||||||
6 | × | 10 | |||||
2 | × | 3 | |||||
2 | × | 6 | |||||
2 Which factors are not included in the prime factorisation of a composite number?
Determine the Prime factorization of numbers:
prime -
Composite - It has more than
Hence,
- 3.And express it in terms of its prime factors.
- The greatest 4-digit number is
. -
is odd check with divisible by 9999x . - sum of digits 9 + 9 + 9 + 9 = 36 , it will divisible by 3.
- sum of digits 3 + 3 + 3 + 3 =
, it will be divisible by 3. - sum of digits 1 + 1 + 1 + 1 = 4 , 4 is not divisible. check with different number
. - 101 is a prime number.
- Thus, the prime factorization of 9999 is:
- 4.And express it in terms of its prime factors.
- The smallest 5-digit number is
. -
is even check with divisible by 10000 x . - This process will be continue until the number reaches divisible by 2.
- 625 is not divisible by 2 check with different number
. - continue until reaches 1.
- Thus, the prime factorization of 10000 is:
5. Find all the prime factors of 1729 and arrange them in ascending order. Now state the relation, if any; between two consecutive prime factors.
check with 1 to 10 numbers which can be divisible.
1729 /
247 is not divisible by 7 then check with 10 to 20.
247 /
19 is a prime number.
Thus, the prime factorization of 1729 is:
Arranging the prime factors in ascending order: 7,13,19.
To find the relationship between two consecutive prime factors: 13 - 7 =
Thus, the difference between each pair of consecutive prime factors is 6. This indicates that the prime factors of 1729 are in an arithmetic progression with a common difference of 6.
6. The product of three consecutive numbers is always divisible by 6. Verify this statement with the help of some examples.
To verify that the product of three consecutive numbers is always divisible by 6, we need to understand that 6 is the product of the prime numbers 2 and 3. Thus, for a product to be divisible by 6, it must be divisible by both 2 and 3.
Example1 : product of
Example1 : product of
Both, this examples are divisible by 2 and 3 then it will also divisible by 6.
7. The sum of two consecutive odd numbers is divisible by 4. Verify this statement with the help of some examples.
To verify that the sum of two consecutive odd numbers is divisible by 4, let's first consider the properties of odd numbers.
Example1 : sum of
Example1 : product of
Both, this examples are divisible by 4.
8. In which of the following expressions, prime factorisation has been done?
24=2x3x4
56=7x2x2x2
70=2x5x7
54=2x3x9
Prime factorization
Not Prime factorization
9. 18 is divisible by both 2 and 3. It is also divisible by 2 × 3 = 6. Similarly, a number is divisible by both 4 and 6. Can we say that the number must also be divisible by 4 × 6 = 24? If not, give an example to justify your answer.
Let us consider the numbers which are divisible by 4 :
Now let us consider the numbers which are divisible by 6 : 6,
So the number divisible by both 4 and 6 are 12,24
Example: 12 /
Thus, 12 is a number that is divisible by both 4 and 6 but not by 4×6=24. Therefore, a number being divisible by both 4 and 6 does not imply that it must be divisible by 24.
10. I am the smallest number, having four different prime factors. Can you find me.
To find the smallest number having four different prime factors, we need to multiply the four smallest prime numbers together. The four smallest prime numbers are
Let calculate the product
Calculating this step-by-step:
So, the smallest number having four different prime factors is:210