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12. I am the smallest number, having four different prime factors. Can you find me?

Since it is the smallest number of such type, it will be the product of 4 smallest prime numbers. that is  2 3  5 7 = 210

10. Determine if 25110 is divisible by 45.
[Hint : 5 and 9 are co-prime numbers. Test the divisibility of the number by 5 and 9].

45 = 5  9 Factors of 5 = 1, 5 Factors of 9 = 1. 3, 9 Therefore, 5 and 9 are co-prime numbers. Since the last digit of 25110 is 0. it is divisible by 5. Sum of the digits of 25110 = 2 + 5 + 1 +1 + 0 = 9 As the sum of the digits of 25110 is divisible by 9, therefore. 25110 is divisible by 9. Since the number is divisible by 5 and 9 both, it is divisible by 45.

9. In which of the following expressions, prime factorisation has been done?
(a) 24 = 2 $\times$ 3 $\times$ 4                    (b) 56 = 7 $\times$ 2 $\times$ 2 $\times$ 2
(c) 70 = 2 $\times$$\times$ 7                    (d) 54 = 2 $\times$ 3 $\times$ 9

in factorization, we don't write composite numbers. All the factors should be prime numbers in this method.  (a) 24 = 2 3 4  As we know that 4 is a composite number.  Hence This is not a prime factorization.  b) 56 = 7 2 2 2 In this factorization, all the factors of 56 are prime numbers. There is no composite number.  Hence This is prime factorization.  c) 70 = 2 5  7  In this...

8. The sum of two consecutive odd numbers is divisible by 4. Verify this statement with the help of some examples.

3+5=8, which is divisible by 4 15+17=32, which is divisible by 4 19+21=40, which is divisible by 4

7. The product of three consecutive numbers is always divisible by 6. Verify this statement with the help of some examples.

2 3 4=24, which is divisible by 6 9 10 11=990, which is divisible by 6 20 21 22=9240, which is divisible by 6

6. Find all the prime factors of 1729 and arrange them in ascending order. Now state the relation, if any; between two consecutive prime factors.

Prime factors of 1729 are -7,13,19. Ascending order =7<13<19. Relation = 7,13,19 differ by 6

5. Write the smallest 5-digit number and express it in the form of its prime factors.

Smallest five-digit number = 10,000 10000 = 2 2 2 2 5 5 5 5

4. Write the greatest 4-digit number and express it in terms of its prime factors.

Greatest four-digit number = 9999 9999 = 3  3 11 101

3. Which factors are not included in the prime factorisation of a composite number?

1 is the factors which are not especially included in the prime factorization of a composite number. When the number is divisible by 2 co-prime numbers then they are divisible by their product.  When two given numbers are divisible by a number, the sum is also divisible by number. When 2 given numbers are divisible by number then its difference is divisible by that number.

2. Here are two different factor trees for 60. Write the missing numbers.
(a)

(a) (b)

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

(a) False as  6 is divisible by 3, but not by 9. (b) True, as 9 = 3 x 3 Therefore, if a number is divisible by 9, then it will also be divisible by 3. (c) False as 30 is divisible by 3 and 6 both, but it is not divisible by 18. (d) True as 9 x 10 = 90 Therefore, If a number is divisible by 9 and 10 both, then it will also be divisible by 90. (e) False as 15 and 32 are co-primes and also...
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