Cyclicity

Question

Let us start this article with a question

Question:

What is the unit digit of \(1234^{567}\)?

To be able to answer the above question, we need to understand what ‘cyclicity’ is and how it is related to unit digits
Let us take the example of 2 and its multiple positive integral power like \(2^1, 2^2, 2^3\) and so on

Attachment:

cyclicity1.png [ 34.35 KiB | Viewed 2499 times ]

If you look closely, you will notice a pattern of 2, 4, 8, and 6 and the pattern repeats itself after every 4th power
- Unit digit of \(2^5\) is the same as the unit digit of \(2^1\)
- Unit digit of \(2^6\) is the same as the unit digit of \(2^2\)
- Unit digit of \(2^7\) is the same as the unit digit of \(2^3\)
- Unit digit of \(2^8\) is the same as the unit digit of \(2^4\)
  - This pattern continues throughout

Does this mean that every number will show the same pattern as 2? 🤔
- Well, let us check one more number. Say for 4

Attachment:

cyclicity2.png [ 22.68 KiB | Viewed 2522 times ]

We see that in the case of 4, the repeating numbers are 4 and 6 and this pattern repeats itself after every 2nd power

Thus, we can say that
- The cyclicity of 2 is 4 (because the pattern 2, 4, 6, and 8 repeats after every 4th power), and
- The cyclicity of 4 is 2 (because the pattern 4 and 6 repeats after every 2nd power)

We can do the same exercise to get the cyclicity of any other digit

Attachment:

cyclicity3.png [ 26.26 KiB | Viewed 2520 times ]

The above table can be shortened and written as

Attachment:

cyclicity4.png [ 20.72 KiB | Viewed 2507 times ]

Trick: This is an easier tale to learn
- Cyclicity of 0, 1, 2, 3, and 4 is the same as cyclicity of 5, 6, 7, 8, and 9 respectively

Answer to the initial question: What is the unit digit of \(1234^{567}\)?

Now coming back to this question, we see that we are looking for the units digit of the number \(1234^{567}\)
We can follow the following steps to get the answer effectively

Step 1: The units digit depends only on the units digit of the base

We can consider this number as \(4^{567}\)

We know 4 has a cyclicity of 2

This suggests \(4^1, 4^3, 4^5\), … will have a units digit of 4
This suggests \(4^2, 4^4, 4^6\), … will have a units digit of 6

Now to understand which pattern this number \(1234^{567}\) will follow we can do the following step

Step 2: We can find out the remainder by dividing the exponent (467) by 2. In this case, the remainder is 1

We can consider the exponent as 1

Step 3: We rewrite the number as \(4^1\) and the answer is \(4\)

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Cyclicity

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Cyclicity

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