Quarter Wit Quarter Wisdom: Cyclicity of Units Digits on the GMAT

Quarter Wit, Quarter WisdomIn our algebra book, we have discussed finding and extrapolating patterns. In this post today, we will look at the patterns we get with various units digits.

The first thing you need to understand is that when we multiply two integers together, the last digit of the result depends only on the last digits of the two integers.

For example:

24 * 12 = 288

Note here: …4 * …2 = …8

So when we are looking at the units digit of the result of an integer raised to a certain exponent, all we need to worry about is the units digit of the integer.

Let’s look at the pattern when the units digit of a number is 2.

Units digit 2:

2^1 = 2

2^2 = 4

2^3 = 8

2^4 = 16

2^5 = 32

2^6 = 64

2^7 = 128

2^8 = 256

2^9 = 512

2^10 = 1024

Note the units digits. Do you see a pattern? 2, 4, 8, 6, 2, 4, 8, 6, 2, 4 … and so on

So what will 2^11 end with? The pattern tells us that two full cycles of 2-4-8-6 will take us to 2^8, and then a new cycle starts at 2^9. 

2-4-8-6

2-4-8-6

2-4

The next digit in the pattern will be 8, which will belong to 2^11. 

In fact, any integer that ends with 2 and is raised to the power 11 will end in 8 because the last digit will depend only on the last digit of the base. 

So 652^(11) will end in 8,1896782^(11) will end in 8, and so on…

A similar pattern exists for all units digits. Let’s find out what the pattern is for the rest of the 9 digits. 

Units digit 3:

3^1 = 3

3^2 = 9

3^3 = 27

3^4 = 81

3^5 = 243

3^6 = 729

The pattern here is 3, 9, 7, 1, 3, 9, 7, 1, and so on…

Units digit 4:

4^1 = 4

4^2 = 16

4^3 = 64

4^4 = 256

The pattern here is 4, 6, 4, 6, 4, 6, and so on… 

Integers ending in digits 0, 1, 5 or 6 have the same units digit (0, 1, 5 or 6 respectively), whatever the positive integer exponent. That is:

1545^23 = ……..5

1650^19 = ……..0

161^28 = ………1

Hope you get the point.

Units digit 7:

7^1 = 7

7^2 = 49

7^3 = 343

7^4 = ….1 (Just multiply the last digit of 343 i.e. 3 by another 7 and you get 21 and hence 1 as the units digit)

7^5 = ….7 (Now multiply 1 from above by 7 to get 7 as the units digit)

7^6 = ….9

The pattern here is 7, 9, 3, 1, 7, 9, 3, 1, and so on…

Units digit 8:

8^1 = 8

8^2 = 64

8^3 = …2

8^4 = …6

8^5 = …8

8^6 = …4

The pattern here is 8, 4, 2, 6, 8, 4, 2, 6, and so on…

Units digit 9: 

9^1 = 9

9^2 = 81

9^3 = 729

9^4 = …1

The pattern here is 9, 1, 9, 1, 9, 1, and so on…

Summing it all up:

1) Digits 2, 3, 7 and 8 have a cyclicity of 4; i.e. the units digit repeats itself every 4 digits.

Cyclicity of 2: 2, 4, 8, 6

Cyclicity of 3: 3, 9, 7, 1

Cyclicity of 7: 7, 9, 3, 1

Cyclicity of 8: 8, 4, 2, 6

2) Digits 4 and 9 have a cyclicity of 2; i.e. the units digit repeats itself every 2 digits.

Cyclicity of 4: 4, 6

Cyclicity of 9: 9, 1 

3) Digits 0, 1, 5 and 6 have a cyclicity of 1.

Cyclicity of 0: 0

Cyclicity of 1: 1

Cyclicity of 5: 5

Cyclicity of 6: 6

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Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!