Step 1: Apply Euclid’s division lemma to 225 and 135
225 = 135 × 1 + 90

Step 2: Now apply to 135 and remainder 90
135 = 90 × 1 + 45

Step 3: Now apply to 90 and remainder 45
90 = 45 × 2 + 0

Since remainder is 0, the HCF is the last non-zero remainder.
∴ HCF(135, 225) = 45

Let a be any positive integer and b = 6. By Euclid’s division lemma:
a = 6q + r, where 0 ≤ r < 6

Possible remainders (r): 0, 1, 2, 3, 4, 5

For odd integers:
– When r = 1: a = 6q + 1 (odd)
– When r = 3: a = 6q + 3 (odd)
– When r = 5: a = 6q + 5 (odd)

r cannot be 0, 2, or 4 as those would make a even (6q, 6q+2, 6q+4).

Hence proved that any odd integer is of form 6q+1, 6q+3 or 6q+5.

Prime factorizations:
12 = 2² × 3
15 = 3 × 5
21 = 3 × 7

HCF = 3 (only common prime factor)
LCM = 2² × 3 × 5 × 7 = 4 × 3 × 5 × 7 = 420

∴ HCF = 3, LCM = 420

Assume √3 is rational. Then it can be written as √3 = a/b where a, b are coprime integers.

Squaring both sides: 3 = a²/b² ⇒ 3b² = a²

This shows a² is divisible by 3 ⇒ a is divisible by 3 (by Fundamental Theorem of Arithmetic).

Let a = 3k. Then 3b² = (3k)² ⇒ 3b² = 9k² ⇒ b² = 3k²

Now b² is divisible by 3 ⇒ b is divisible by 3.

But this contradicts our assumption that a and b are coprime (both divisible by 3).

Hence, √3 is irrational.

For a number to end with 0, its prime factorization must contain both 2 and 5.

6ⁿ = (2 × 3)ⁿ = 2ⁿ × 3ⁿ

The prime factors are only 2 and 3. There is no factor of 5.

∴ 6ⁿ cannot end with digit 0 for any natural number n.

Using the relationship: HCF × LCM = Product of two numbers

9 × LCM = 306 × 657

LCM = (306 × 657) / 9

LCM = 34 × 657 = 22,338

∴ LCM(306, 657) = 22,338

7 × 11 × 13 + 13 = 13(7 × 11 + 1) = 13 × 78 = 13 × 2 × 3 × 13

The expression has prime factors other than 1 and itself.

By definition, a composite number has more than two distinct prime factors.

Hence, it is a composite number.

Assume 3 + 2√5 is rational. Then it can be written as p/q where p, q are integers.

3 + 2√5 = p/q ⇒ 2√5 = (p/q) – 3 ⇒ √5 = (p – 3q)/(2q)

This implies √5 is rational (since right side is rational).

But we know √5 is irrational. This is a contradiction.

Hence, 3 + 2√5 is irrational.

This problem requires finding the LCM of their time taken.

Prime factorizations:
12 = 2² × 3
18 = 2 × 3²

LCM = 2² × 3² = 4 × 9 = 36 minutes

∴ They will meet after 36 minutes.

Using factor tree method:

156 ÷ 2 = 78
78 ÷ 2 = 39
39 ÷ 3 = 13

156 = 2 × 2 × 3 × 13 = 2² × 3 × 13

∴ Prime factorization: 156 = 2² × 3 × 13