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Question : 

1) Prove that every positive integer different from 1 can be expressed as a product of a non-negative power of 2 and an odd number.

Doubt by Vanshika

Explanation : 

Let us understand this with an example : 

Let us take any positive integer which is different from 1

Case I

10 = 2×5

Clearly 10 is expressed as a product of non negative power of 2 and an odd number.

Case II

20 = 2²×5

Clearly 20 is expressed as a product of non negative power of 2 and an odd number.

Case III

30 = 2¹×3¹×5¹ = 2¹×15

Clearly 30 is expressed as a product of non negative power of 2 and an odd number.

Case IV

31 = 2⁰×31

Clearly 31 is expressed as a product of non negative power of 2 and an odd number.

I hope you got it. 

2) Prove that a positive integer n is prime, if no prime p less than or equal to √n divides n.

Explanation : 

Let us take a positive integer n

Case I

n=19

Now √19 = 4.35

Now Prime Number less than or equal to 4.35 are 2 and 3 and we can clearly see that out of 2 and 3 no one completely divides n i.e. 19. Hence we can say that 19 is a prime number.

Case II

n = 59 

Now √59 = 7.68

Now Prime Number less than or equal to 7.68 are 2, 3, 5 and 7 and we can again clearly see that out of 2, 3, 5 and 7 no one completely divided n i.e. 59. Hence we can say that 59 is a prime number.

Case III

n = 65

Now √65 = 8.06

Now Prime Number less than or equal to 8.06 are 2, 3, 5 and 7 but this time we can see that out of 2, 3, 5 and 7, the number 5 completely divided n i.e. 65. Hence we can conclude that 65 is not a prime number.

I hope you got it.