# CBSE Class 10-Mathematics: Chapter –1 Real Numbers Part 1

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**1 Marks Questions**

**Question 1:** Use Euclid Division lemma to show that the square of any positive integer is either of form for some integer

[Hint: Let x be any positive integer then it is of the form . Now square each of these and show that they can be rewritten in the form .]

**Answer:**

Let a be any positive integer and .

The for some integer

And because

Therefore, or

Or,

Where are some positive integers?

Hence, it can be said the square of any positive integer is either of the form .

**Question 2:**

Express each number as product of its prime factors:

(i)

(i)

(iv) 5005

(v)

**Answer:**

(i)

(ii)

(iii)

(iv)

**Question 3:**

Given that , find

**Answer:**

We know that Product of two numbers

**Question 4:**

Check whether can end with the digit for any natural number

**Answer:**

If any number ends with the digit , it should be divisible by .

In other words, it will also be divisible by and as

Prime Factorization of

It can be observed that is not in the prime factorisation of

Hence, for any value of will not be divisible by .

Therefore, cannot end with the digit for any natural number .

**Question 5:**

Prove that is Irrational

**Answer:**

Let us say is rational.

Then the co-prime x and y of the given rational number where is such that:

Rearranging, we get,

Since x and y are integers, thus, is a rational number.

Therefore, is also a rational number. But this confronts the fact that is irrational.

Hence, we get that is irrational.

**Question 6:**

is a

(a) Composite number

(b) Whole number

(c) Prime number

(d) None of these

**Answer:**

(a) and (b) both