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"20. 23, 20, এবং 31এর গ.সা.গু বের করো। 5 এবং 6 এর ল.সা.গু বের করো। দুটি জ্যামিতিক শ্রেণির প্রথম পদ 12 এবং সাধারণ অনুপাত 2 হলে প্রথম 3 পদ নির্ণয় করো। একটি আয়তাকার ঘরের দৈর্ঘ্য 3.5 মিটার এবং প্রস্থ 2.3 মিটার হলে ঘরের ক্ষেত্রফল নির্ণয় করো।"

This translates to the following mathematical tasks:

1. Find the GCD (Greatest Common Divisor) of 23, 20, and 31.
2. Find the LCM (Least Common Multiple) of 5 and 6.
3. For a geometric sequence with the first term 12 and a common ratio of 2, find the first 3 terms.
4. Calculate the area of a rectangular room with a length of 3.5 meters and a width of 2.3 meters.

Let's solve each of these problems.

1. GCD of 23, 20, and 31

The GCD of 23, 20, and 31 is 1 since these numbers are relatively prime (they do not have any common factors other than 1).

2. LCM of 5 and 6

The LCM of 5 and 6 is 30 because the smallest multiple of both 5 and 6 is 30.

3. First 3 terms of the geometric sequence

For a geometric sequence with a first term $a_1 = 12$ and common ratio $r = 2$, the first three terms are:

• $a_1 = 12$
• $a_2 = 12 \times 2 = 24$
• $a_3 = 24 \times 2 = 48$

So, the first three terms are 12, 24, 48.

4. Area of the rectangular room

The area $A$ of a rectangle is calculated by the formula: $A = \text{Length} \times \text{Width}$ Substituting the values: $A = 3.5 \times 2.3 = 8.05 \text{ square meters}$

So, the area of the room is 8.05 square meters.

Would you like more details on any of these solutions or have any additional questions?

Here are five related questions for further practice:

1. What is the GCD of 15, 25, and 35?
2. Find the LCM of 8 and 12.
3. If a geometric sequence has a first term of 5 and a common ratio of 3, what are the first 4 terms?
4. How would you find the area of a rectangular room if its length is given in feet and width in meters?
5. What is the GCD of 44, 88, and 132?

Tip: When working with GCD and LCM, remember that the GCD focuses on the common factors of numbers, while the LCM focuses on common multiples.