Unit Digits using Cyclicity

Let us start this article with a question

Question:

What is the unit digit of 1234⁵⁶⁷?

• 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 2¹, 2², 2³ 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 2⁵ is the same as the unit digit of 2¹
  • The unit digit of 2⁶ is the same as the unit digit of 2²
  • The unit digit of 2⁷ is the same as the unit digit of 2³
  • The unit digit of 2⁸ is the same as the unit digit of 2⁴
    • 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 1234⁵⁶⁷?

Now coming back to this question, we see that we are looking for the units digit of the number 1234⁵⁶⁷

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 4⁵⁶⁷

We know 4 has a cyclicity of 2

• This suggests 4¹, 4³, 4⁵, … will have a units digit of 4
• This suggests 4², 4⁴, 4⁶, … will have a units digit of 6

Now to understand which pattern this number 1234⁵⁶⁷ 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 4¹ and the answer is 4