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Question 1.
A positive integer n when divided by 9, gives 7 as remainder. Find the remainder when (3n – 1) is divided by 9.
Question 1.
A positive integer n when divided by 9, gives 7 as remainder. Find the remainder when (3...
Solution:
Here n can be written as 9k + 7, where k ∈ N
Now 3n – 1 = 3(9k + 7) – 1 = 27k + 20
Applying Euclid’s division lemma on (27k + 20) and 9, we have
27k + 20 = 9 × (3k + 2) + 2;
where k ∈ N
Thus, 2 is the remainder.
Q.3: Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:
(i) 13/3125 (ii) 17/8 (iii) 64/455 (iv) 15/1600
Q.3: Without actually performing the long division, state whether the following rational numbers wil...
Note: If the denominator has only factors of 2 and 5 or in the form of 2m × 5n then it has a terminating decimal expansion.
If the denominator has factors other than 2 and 5 then it has a non-terminating repeating decimal expansion.

(i) 13/3125

Factoring the denominator, we get,

3125 = 5 × 5 × 5 × 5 × 5 = 55

Or

= 20 × 55

Since the denominator is of the form 2m × 5n then, 13/3125 has a terminating decimal expansion.

(ii) 17/8

Factoring the denominator, we get,

8 = 2× 2 × 2 = 23

Or

= = 23 × 50

Since the denominator is of the form 2m × 5n then, 17/8 has a terminating decimal expansion.

(iii) 64/455

Factoring the denominator, we get,

455 = 5 × 7 × 13

Since the denominator is not in the form of 2m × 5n, therefore 64/455 has a non-terminating repeating decimal expansion.

(iv) 15/1600

Factoring the denominator, we get,

1600 = 26 × 52

Since the denominator is in the form of 2m × 5n, 15/1600 has a terminating decimal expansion.
Q.4: Prove that 3 + 2√5 is irrational.
Q.4: Prove that 3 + 2√5 is irrational.
Solution:

Let 3 + 2√5 be a rational number.

Then the co-primes x and y of the given rational number where (y ≠ 0) is such that:

3 + 2√5 = x/y

Rearranging, we get,

2√5 = (x/y) – 3

√5 = 1/2[(x/y) – 3]

Since x and y are integers, thus, 1/2[(x/y) – 3] is a rational number.

Therefore, √5 is also a rational number. But this confronts the fact that √5 is irrational.

Thus, our assumption that 3 + 2√5 is a rational number is wrong.

Hence, 3 + 2√5 is irrational.