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

- If you look closely, you will notice a pattern of 2, 4, 8, and 6 and the pattern repeats itself after every 4
^{th} power
- The unit digit of 2
^{5} is the same as the unit digit of 2^{1}
- The unit digit of 2
^{6} is the same as the unit digit of 2^{2}
- The unit digit of 2
^{7} is the same as the unit digit of 2^{3}
- The 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

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

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

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

- The above table can be shortened and written as

- Trick: This is an easier tale to learn
- Cyclicity of 0, 1, 2, 3, and 4 is the same as 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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Let us start this article with a question

Question:What is the unit digit of 1234^{567}?^{1}, 2^{2}, 2^{3}and so on^{th}power^{5}is the same as the unit digit of 2^{1}^{6}is the same as the unit digit of 2^{2}^{7}is the same as the unit digit of 2^{3}^{8}is the same as the unit digit of 2^{4}^{nd}power^{th}power) and^{nd}power)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^{567}We know 4 has a cyclicity of 2

^{1}, 4^{3}, 4^{5}, … will have a units digit of 4^{2}, 4^{4}, 4^{6}, … will have a units digit of 6Now to understand which pattern this number

1234will follow we can do the following step^{567}Step 2:We can find out the remainder by dividing the exponent (467) by 2. In this case, the remainder is 1Step 3:We rewrite the number as 4^{1}and the answer is 4## Related posts:

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