Ex 3.5 Question 1.
Which of the following statements are true?
(a) If a number is divisible by 3, it must be divisible by 9.
(b) If a number is divisible by 9, it must be divisible by 3.
(c) A number is divisible by 18, if it is divisible by both 3 and 6.
(d) If a number is divisible by 9 and 10 both, then it must be divisible by 90.
(e) If two numbers are co-primes, at least one of them must be prime.
(f) All numbers which are divisible by 4 must also be divisible by 8.
(g) All numbers which are divisible by 8 must also be divisible by 4.
(h) If a number exactly divides two numbers separately,- it must exactly divide their sum.
(i) If a number exactly divides the sum of two numbers, it must exactly divide the two numbers separately.
Solution:
(a) False
(b) True
(c) False
(d) True
(e) False
(f) False
(g) True
(h) True
(i) False
Ex 3.4 Question 2.
Here are two different factor trees for 60. Write the missing numbers.
Solution:
(a)
(b) 
Ex 3.4 Question 3.
Which factors are not included in the prime factorization of a composite number?
Solution:
1 is the factor that is not included in the prime factorization of a composite number.
Ex 3.4 Question 4.
Write the greatest 4-digit number and express it in terms of its prime factors.
Solution:
The greatest 4-digit number = 9999
Hence, the prime factors of 9999 = 3 x 3 x 11 x 101.
Ex 3.4 Question 5.
Write the smallest 5-digit number and express it in the form of its prime factors.
Solution:
The smallest 5-digit number = 10000
Hence, the required prime factors: 10000 = 2 x 2 x 2 x 2 x 5 x 5 x 5 x 5.
Ex 3.4 Question 6.
Find all the prime factors of 1729 and arrange them in ascending order. Now state the relations, if any, between the two consecutive prime factors.
Solution:
Hence, the prime factors of 1729 = 7 x 13 x 19.
Here, 13 – 7 = 6 and 19 – 13 = 6
The difference between two consecutive prime factors is 6.
Ex 3.4 Question 7.
The product of three consecutive numbers is always divisible by 6. Verify this statement with the help of some examples.
Solution:
Example 1:
Take three consecutive numbers 2, 3 and 4.
2 x 3 x 4 = 24
Therefore, the product 2 x 3 x 4 = 24 is divisible by 6.
Example 2:
Take three consecutive numbers 4 ,5 and 6.
4 x 5 x 6 = 120
Therefore, the product 4 x 5 x 6 = 120 is divisible by 6.
Ex 3.4 Question 8.
The sum of two consecutive odd numbers is divisible by 4. Verify this statement with the help of some examples.
Solution:
Example 1:
5 + 3 = 8 and 8 is divisible by 4.
Example 2:
7 + 5 = 12 and 12 is divisible by 4.
Example 3:
9 + 7 = 16 and 16 is divisible by 4.
Ex 3.4 Question 9.
In which of the following expressions, prime factorisation has been done?
(a) 24 = 2 x 3 x 4
(b) 56 = 7 x 2 x 2 x 2
(c) 70 = 2 x 5 x 7
(d) 54 = 2 x 3 x 9.
Solution:
(a) 24 = 2 x 3 x 4
Here, 4 is not a prime number.
Hence, 24 = 2 x 3 x 4 is not a prime factorisation.
(b) 56 = 7 x 2 x 2 x 2
Here, all factors are prime numbers
Hence, 56 = 7 x 2 x 2 x 2 is a prime factorisation.
(c) 70 = 2 x 5 x 7
Here, all factors are prime numbers.
Hence, 70 = 2 x 5 x 7 is a prime factorisation.
(d) 54 = 2 x 3 x 9
Here, 9 is not a prime number.
Hence, 54 = 2 x 3 x 9 is not a prime factorisation.
Ex 3.4 Question 10.
Determine if 25110 is divisible by 45.
Solution:
45 = 5 x 9
Here, 5 and 9 are co-prime numbers.
Test of divisibility by 5: a unit place of the given number 25110 is 0. So, it is divisible by 5.
Test of divisibility by 9:
Sum of the digits = 2 + 5 + l + l + 0 = 9 which is divisible by 9.
So, the given number is divisible by 5 and 9 both. Hence, the number 25110 is divisible by 45.
Ex 3.4 Question 11.
18 is divisible by both 2 and 3. It is also divisible by 2 x 3 = 6. Similarly, a number is divisible by both 4 and 6. Can we say that the number must also be divisible by 4 x 6 = 24? If not, give an example to justify your answer.
Solution:
The given two numbers are not co-prime. So, it is not necessary that a number divisible by both 4 and 6, must also be divisible by their product 4 x 6 = 24.
Ex 3.4 Question 12.
I am the smallest number, having four different prime factors. Can you find me?
Solution:
The smallest 4 prime numbers = 2, 3, 5, and 7.
Hence, the required number = 2 x 3 x 5 x 7 = 210
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