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**Ques 1: Show that 6 ^{n} cannot end with the digit 0.**

âˆµ All composite numbers ending with 0 have 10 as a factor.

âˆ´10 is a factor of 6

âˆ´ We have: 6

â‡’ (2 Ã— 3)

â‡’ 2

i.e. 5 is a prime factor of 2

Thus, our assumption was wrong and 6

If possible, let â€˜nâ€™ be a positive integer such that is rational.

Let

(using componendo and dividendo)

Since are rational numbers.

âˆ´ are also rational.

i.e. are perfect squares of +ve integers. It is not possible.

[âˆµ two perfect squares differ by 3] â‡’ Our assumption is not possible.**Ques 3: If is an irrational number, then show that is also irrational. ****Hint:**

If possible, let be a rational number.

But it contradicts our assumption thatis irrational.Thus, is irrational.

**Ques 4: How many prime numbers are there of the form 10n + 3, where n is a whole number 1 < n < 10?**

Substituting 1, 2, 3, .................... 10 in 10 n + 3, we have:

Thus, the required numbers are: 13, 23, 43, 53, 73 and 83.

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