Last digit of a power

Note: While multiplying two multiple digit numbers, the last digit of the product is the last digit of the product of the last digit of the two numbers. For example, 332 × 332 = 110224, the last digit of this product can be simply obtained by multiplying 2 × 2 = 4. The last digit of a number ab forms a particular sequence or order depending on the unit digit of the number (a) and the power the number is raised to (b) This sequence can be called the power cycle of a number and thus it depends on its unit digit. Consider the power cycle of 2:

2¹ = 2, 2² = 4,

2³ = 8,

Clearly, we can observe that the unit digit gets repeated after

Hence, we can say that 2 has a power cycle of 2, 4, 6, 8, with frequency 4.

Thus, every power of 2.

The numbers of the form 2+1 have last digit 2.

The numbers of the form 2⁴+2 have last digit 4.

The numbers of the form 2⁴+3 have last digit 8.

The numbers of the form 2⁴+4 have last digit 6.

where, k = 0, 1, 2, 3, ...

Note that

4k+ 1 means the set of natural numbers, which gives remainder 1 when divided by 4.

4k + 2 means the set of natural numbers, which gives remainder 2 when divided by 4.

4k + 3 means the set of natural numbers, which gives remainder 3 when divided by 4.

4k + 4 means the set of natural numbers, which gives remainder 0 when divided by 4.

Note: For all the numbers ending in 2, the power cycle and its frequency is the same as that of 2.