# CBSE Class 10-Mathematics: Chapter – 1 Real Numbers Part 3 (For CBSE, ICSE, IAS, NET, NRA 2022)

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**Question 15**:

is

(a) An integer

(b) An irrational number

(c) A rational number

(d) None of these

**Answer**:

(c) A rational number

**Question 16**:

is

(a) A rational number

(b) An irrational number

(c) Both (a) & (b)

(d) Neither rational nor irrational

**Answer**:

(b) An irrational number

**Question 17**:

is

(a) A rational number

(b) An irrational number

(c) An integer

(d) Not real number

**Answer**:

(b) An irrational number

## 2 Mark Questions

**Question 1**:

Show that any positive odd integer is of the form , or , where is some integer.

**Answer**:

Let be any positive integer and . Then, by Euclid՚s algorithm,

for some integer , and because .

Therefore, or or

Also, , where is a positive integer

, where is an integer

, where is an integer

Clearly, are of the form , where is an integer.

Therefore, are not exactly divisible by . Hence, these expressions of numbers are odd numbers.

And therefore, any odd integer can be expressed in the form , or ,

Or

**Question 2**:

An army contingent of members is to march behind an army band of members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

**Answer**:

We have to find the to find the maximum number of columns in which they can march.

To find the , we can use Euclid՚s algorithm.

The is .

Therefore, they can march in 8 columns each.

**Question 3**:

Use Euclid՚s division lemma to show that the cube of any positive integer is of the form .

**Answer**:

Let *a* be any positive integer and

, where and

Therefore, every number can be represented as these three forms.

We have three cases.

**Case 1**: When

Where *m* is an integer such that

**Case 2**: When ,

Where is an integer such that

**Case 3**: When ,

Where *m* is an integer such that

Therefore, the cube of any positive integer is of the form , or .