Read the data carefully and answer the following question.

Given the sequence: $$20, a, b, c, d, 65$$

where,

- $$a - 20 = X^2 + Y$$
- $$b - a = (X + 1)^2 + Y$$ (with $$b a$$)
- $$c - b = (X + 2)^2 + Y$$ (with $$c b$$)
- $$d - c = (X + 3)^2 + Y$$ (with $$d c$$)

Here,

- $$X$$ is the HCF of two prime numbers $$M$$ and $$N$$
- $$Y$$ is the smaller root of the equation $$z^2 - z - 6 = 0$$

**Question:** How many of the numbers $$a, b, c, d$$ are divisible by 3?

Question

Read the data carefully and answer the following question.

Given the sequence: $$20, a, b, c, d, 65$$

where,

- $$a - 20 = X^2 + Y$$
- $$b - a = (X + 1)^2 + Y$$ (with $$b > a$$)
- $$c - b = (X + 2)^2 + Y$$ (with $$c > b$$)
- $$d - c = (X + 3)^2 + Y$$ (with $$d > c$$)

Here,

- $$X$$ is the HCF of two prime numbers $$M$$ and $$N$$
- $$Y$$ is the smaller root of the equation $$z^2 - z - 6 = 0$$

**Question:** How many of the numbers $$a, b, c, d$$ are divisible by 3?

Accepted Answer

**Solution:**

1. First, find the values of *X* and *Y*.

- *X* is the HCF of two prime numbers *M* and *N*. Since both are prime, their HCF can only be 1 (if they are different primes) or the prime itself (if they are the same). But since they are two primes, the only common factor is 1. So, $$X = 1$$.

- *Y* is the smaller root of the quadratic equation:
  $$z^2 - z - 6 = 0$$

Solve using the quadratic formula:
  $$z = \frac{1 \pm \sqrt{1 + 24}}{2} = \frac{1 \pm 5}{2}$$

Roots are:
  - $$\frac{1 + 5}{2} = 3$$
  - $$\frac{1 - 5}{2} = -2$$

The smaller root is $$Y = -2$$.

2. Calculate each term:

- $$a - 20 = X^2 + Y = 1^2 + (-2) = 1 - 2 = -1$$
  $$\Rightarrow a = 20 - 1 = 19$$

- $$b - a = (X + 1)^2 + Y = (1 + 1)^2 + (-2) = 2^2 - 2 = 4 - 2 = 2$$
  $$\Rightarrow b = a + 2 = 19 + 2 = 21$$

- $$c - b = (X + 2)^2 + Y = (1 + 2)^2 - 2 = 3^2 - 2 = 9 - 2 = 7$$
  $$\Rightarrow c = b + 7 = 21 + 7 = 28$$

- $$d - c = (X + 3)^2 + Y = (1 + 3)^2 - 2 = 4^2 - 2 = 16 - 2 = 14$$
  $$\Rightarrow d = c + 14 = 28 + 14 = 42$$

3. Now check divisibility by 3:

- $$a = 19$$ → Not divisible by 3
- $$b = 21$$ → Divisible by 3
- $$c = 28$$ → Not divisible by 3
- $$d = 42$$ → Divisible by 3

4. The question asks how many numbers are divisible by 3 among $$a, b, c, d$$.

- Numbers divisible by 3 are $$b$$ and $$d$$.

**Answer:** Option (c) **b and d only**

**Note:** The user selected option (c) in the input, but the calculation shows that only *b* and *d* are divisible by 3, so the correct answer is option (c).

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