Units digit of a number raised to power

Here, we will see how to find the units digit of a number that is in the form ${x^y}$. We will first try to understand what is a units digit, then we will look at the technique to find the units digit of large powers and then using this technique we will solve some problems on Units digit of a number raised to power.

At the end, please take the QUIZ to test your understanding.

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What is a Units digit?

Units digit of a number is the digit in the one’s place of the number. i.e It is the rightmost digit of the number. For example, the units digit of 243 is 3, the units digit of 39 is 9.

But then what is the units digit of large numbers like 23 to the power 46 or what is the units digit of 2014 to the power of 2014? Here, it is not straight forward to calculate the units digit of these numbers. So lets have a look at the technique to calculate the units digit of large numbers.

Units digit of Large Numbers – number raised to power

One of the ways of finding the units digit of a power is by finding the remainder when that number is divided by 10.
Another general and one of the easier ways to find the units digit of a number in the form ${x^y}$, is done with the help of the following steps:

  1. Identify the units digit in the base ‘x’ and call it say ‘l’. {For example, If x = 24, then the units digit in 24 is 4. Hence l = 4.}
  2. Divide the exponent ‘y’ by 4.
    • If the exponent y is exactly divisible by 4. i.e, y leaves a remainder 0 when divided by 4. Then,
      • the units digit of ${x^y}$ is 6, if l = 2, 4, 6, 8.
      • the units digit of ${x^y}$ is 1, if l = 3, 7, 9.
    • If y leaves a non-zero remainder r, when divided by 4 (i.e y = 4k + r). Then,
      • the units digit of ${x^y}$ = ${l^r}$

Problems on Units digit of large powers

Example 1: what is the units digit of 2014 to the power of 2012?

Here, we have to find the units digit of ${2014^{2012}}$

  1. The base is 2014 and hence its units digit is 4. Therefore, l = 4.
  2. The exponent is 2012, which is divisible by 4.
    • Since l is even and the exponent is divisible by 4, we have the units digit of 2014 to the power of 2012 is 6.

Example 2: what is the units digit in the expansion 1453 raised to 71?

Here, we have to find the units digit of ${1453^{71}}$

  1. The base is 1453 and hence its units digit is 3. Therefore, l = 3.
  2. The exponent is 71, which when divided by 4 gives a remainder 3.
      • Since l is 3 and the exponent leaves a remainder 3 when divided by 4, we have

    the units digit of ${1453^{71}}$ = the units digit of ${3^3}$ = 7

Example 3. what is the units digit in the expansion of 2 the power of 51?

Here, we have to find the units digit of ${{2^{51}}}$

  1. Since the base is 2. The units digit in base, i.e  l = 2.
  2. Now the exponent 51 when divided by 4 leaves remainder 3. Hence, the units digit of 2 to the power of 51 is given by units digit of 2 to the power of 3 which is 8.


Quiz : Test Your Understanding

We hope that you now have a good idea on how to find the units digit of a number raised to power.  Here is a small quiz to test your understanding.

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