Prime Factorisation

Q.

Prove that for any positive integer $n,n^3-n$ is divisible by $6$.

Q.

Express 5005 as a product of its prime factors.

Q.

If 3 is is the least prime factor of number a and 7 is the least prime factor of number b, then the least prime factor of a+b , is

Q. a and b are two positive integers such that the least prime factor of a is 3 and and that of b is 5.find the least prime factor of (a+b).

Q.

Find the simplest form of :

$\left(i\right)\frac{69}{92}\left(ii\right)\frac{473}{645}\left(iii\right)\frac{1095}{1168}\left(iv\right)\frac{368}{496}$

Q.

Whether $8$power $n$ , where $n$ is a natural number, can end with zero?

Q.

Express 140 as the product of its prime factors:

Q.

Determine the prime factorisation of each of the following positive integer:
(i) 20570 (ii) 58500 (iii) 45470971

Q.

Which of the following are the prime factors of 5005?

1. 2

2. 3

3. 4

4. 5

Q.

Express each of the following integers as a product of its prime factors:
(i) 420 (ii) 468 (iii) 945 (iv) 7325

Q.

Express 360 as product of its prime factors.

Q.

Question 1 (ii)
Express 156 as the product of its prime factors.

Q. The number of prime factors of 36 is:

1. 2
2. 3
3. 4
4. 6

Q. What is the total number of factors of a prime number?

Q. The sum of two prime number is always a prime number (True/ False).

Q.

How many prime factors are there in prime factorisation of 5005?

1. 7

2. 6

3. 2

4. 4

Q. Express 23150 as product of its prime factors. Is it unique?

Q.

A boy with a collection of marble realise that if he makes a group of 5 or 6 marbles at a time there are always 2 marbles left. Can you explain why the boy cannot have a prime number of marbles.

Q.

The simplest form of $\frac{1095}{1168}$ is

$\left(a\right)\frac{17}{26}\left(b\right)\frac{25}{26}\left(c\right)\frac{13}{16}\left(d\right)\frac{15}{16}$

Q.

Express 7429 as a product of its prime factors.

Q.

Why is 1 neither prime nor composite numbers?

Q.

Is $0$ an even number?

Q.

Question 27
The number of distinct prime factors of the largest 4-digit number is

(A) 2
(B) 3
(C) 5
(D) 11

Q. The prime factorisation of 272 is .

1. $2^2\times 17$
2. $2^4\times 17$
3. $2^4\times 13$

Q.

Find the common factors of:

$35$ and $50$

Q.

Will division of a rational number by another rational number always give a rational number?

Q.

The sum of prime factors of 4620 is:

1. 28

2. 32

3. 30

4. 34

Q.

Question 5 (c)

Which of the following numbers are co-prime:
30 and 415

Q. Match the following:

1.Total number of factors of 360	i) 78
2. Sum of all factors of 360	ii) 1092
3.Sum of all odd factors of 360	iii) 24
4.Sum of all even factors of 360	iv) 1170
5.Product of all factors of 360	v) ${360}^{12}$

1. 1-v, 2-i, 3-iv, 4-iii, 5-ii
2. 1-iii, 2-iv, 3-i, 4-ii, 5-v
3. 1-ii, 2-v, 3-i, 4-iii, 5-iv
4. 1-i, 2-iv, 3-iii, 4-ii, 5-v

Q.

'a' and 'b' are two positive integers such that the least prime factor of 'a' is 3 and the least prime factor of 'b' is 5.Then find the least prime factor of (a+b).