Question

Write the sequence of natural numbers which leaves a remainder 1 on division by 4.

Hint:

To start such a sequence, find the smallest natural number that fits the condition. Here, it's 1. Then, keep adding the divisor (in this case, 4) to get the subsequent terms.

Answer 1

We need to list the natural numbers that, when divided by 4, give a remainder of 1.

A number that leaves a remainder of 1 when divided by 4 can be expressed in the form 4k + 1, where k is a non-negative integer (0, 1, 2, ...).

Let's find the first few terms of the sequence by substituting values for k:
- If k = 0, the number is 4(0) + 1 = 1. (1 divided by 4 is 0 with a remainder of 1).
- If k = 1, the number is 4(1) + 1 = 5. (5 divided by 4 is 1 with a remainder of 1).
- If k = 2, the number is 4(2) + 1 = 9. (9 divided by 4 is 2 with a remainder of 1).
- If k = 3, the number is 4(3) + 1 = 13. (13 divided by 4 is 3 with a remainder of 1).
The sequence continues in this pattern.

The sequence is 1, 5, 9, 13, ....