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 4th 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 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
  • 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

Alligation Made Easy

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