Let's go through this step-by-step, using each clue provided to deduce the six numbers.

Information Given

1. The lowest number is 37.
2. The range is 24.
   • This means the difference between the largest and smallest numbers is 24.
   • Therefore, the largest number is $37 + 24 = 61$.
3. The mode is 43.
   • This means 43 is the most frequent number, so it must appear more than once.
4. The median is 46.
   • With six numbers, the median is the average of the third and fourth numbers when sorted in ascending order.
   • This suggests that the third and fourth numbers average to 46, so they could both be 46.
5. One number is a multiple of 11.
   • Multiples of 11 near the given range (37 to 61) include 44 and 55. One of these must be in the list.

Step-by-Step Solution

1. List Known Numbers:

   • We know the smallest number is 37.
   • The largest number is 61.

2. Assign 43 as the Mode:

   • Since 43 is the mode, it must appear more than once. Let’s assume there are two occurrences of 43 for now.

3. Median Requirement:

   • The median is 46, which suggests the third and fourth numbers in ascending order are 46. So, we can assign two positions with the value 46.

4. Possible Numbers So Far:

   • From the information we’ve assumed, the six numbers could look something like this (in ascending order): $37, 43, 46, 46, \text{(another number)}, 61$
   • We still need one more number, which must be a multiple of 11. The two possible multiples of 11 are 44 and 55.

5. Inserting a Multiple of 11:

   • If we use 44 as the fifth number, we would get the following sorted set: $37, 43, 44, 46, 46, 61$
   • This arrangement satisfies all conditions:
     • Lowest number is 37.
     • Range is $61 - 37 = 24$.
     • Mode is 43 (since 43 appears twice).
     • Median is 46 (average of the third and fourth numbers, 44 and 46).
     • One number is a multiple of 11 (44).

Conclusion

The six numbers are: $37, 43, 43, 46, 46, 61$

Let me know if you'd like further clarification on any of these steps!

Would you like to explore any of these follow-up questions?

1. Why can’t 55 be the multiple of 11 in this case?
2. What would happen if there was only one occurrence of 43?
3. How would changing the median affect the solution?
4. Can the mode be a number other than 43?
5. How does the range help narrow down possible answers?

Tip: Always arrange numbers in ascending order to make identifying the median and range easier.