A is the set of the first 100 natural numbers. What is the minimum numbe of elements that should be picked from A to ensure that atleast one pair of numbers whose difference is 10 is picked?

Options

A. 51
B. 55
C. 20
D. 11

Answer

To solve the problem, we need to determine the minimum number of elements that must be picked from the set A (the first 100 natural numbers) to ensure that at least one pair of numbers has a difference of 10.

### Step-by-Step Solution:

1. **Understanding the Set A**:
The set A consists of the first 100 natural numbers:
\[ A = \{ 1, 2, 3, \ldots, 100 \} \]

2. **Identifying Pairs with a Difference of 10**:
We need to find pairs of numbers in the set A such that their difference is 10. The pairs can be identified as follows:
- (1, 11)
- (2, 12)
- (3, 13)
- (4, 14)
- (5, 15)
- (6, 16)
- (7, 17)
- (8, 18)
- (9, 19)
- (10, 20)
- (11, 21)
- (12, 22)
- ...
- (90, 100)

In total, there are 10 such pairs: (1, 11), (2, 12), ..., (10, 20), (11, 21), ..., (90, 100).

3. **Choosing Numbers to Avoid Pairs**:
To avoid selecting any pairs whose difference is 10, we can choose one number from each pair. For example, we can choose:
- From (1, 11), we can choose 1.
- From (2, 12), we can choose 2.
- ...
- From (10, 20), we can choose 10.

This way, we can select:
\[ \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \} \]
This gives us 10 numbers.

4. **Maximizing Selections Without Forming Pairs**:
Continuing this process, we can select:
- From (11, 21), we can choose 11.
- From (12, 22), we can choose 12.
- ...
- From (90, 100), we can choose 90.

If we continue this pattern, we can select a total of 50 numbers without forming any pairs with a difference of 10.

5. **Ensuring At Least One Pair**:
If we select 50 numbers, we can avoid forming any pairs with a difference of 10. However, when we select the 51st number, it must be one of the numbers that we have not yet chosen, which will inevitably create at least one pair with a difference of 10.

### Conclusion:
Thus, the minimum number of elements that should be picked from A to ensure that at least one pair of numbers whose difference is 10 is picked is **51**.