Finding Last Digit of a Number

Concepts Examples

To find the last digit i.e the unit digit you need to be aware of the cyclicity of the numbers and their powers

21=2	31=3	41=4
22=4	32=9	42=16
23=8	33=27	43=64
24=16	34=81	44=256
25=32	35=243	45=1024
26=64	36=729	46=4096
27=128	37=2187
28=256	38=6561

We can see that unit's digit of 2¹,2⁵,2⁹ is 2 and so on. Therefore, after every four powers of 2, the units digit of the number starts repeating.Thus we can say that cyclicity of unit's digit of higher powers of 2 is 4.

Similarily unit digit of power of 4 starts reeating after 2, thus its cyclicity is 2.

Unit digit follows a periodic pattern that is after a particular period it repeats in a cyclic form, this is called cyclicity.

• Unit digits of numbers ending with 0,1,5,6 is always the same irrespective of their powers raised on them.

• Unit digit of numbers ending with 4,9 follows the pattern with cyclicity of 2

• Unit digit of numbers ending with 2,3,7,8 follows the pattern with cyclicity of 4

In order to find the last digit of any number xyz^abc whose who's last digit 'z' has the cyclicity of 4, then to find the last digit write the number as z^{4k + m} where 4k+m = abc, and m is smaller than equal to z and not 0.

In order to find the last digit of any number xyz^abc whose who's last digit 'z' has the cyclicity of 2, then to find the last digit write the number as z^{2k + m} where 2k+m = abc, and m is smaller than or equal to z ans is not equal to 0.

Question 1 What will be the unit digit of 7²³×8¹³

2	3
4	6

C

Since the cyclicity of both 7 and 8 is 4,

∴ unit digit of 7²³ and 8¹³=7^4×5+3 × 8^4×3+1 =unit digit of 7³ and 8¹

∴ unit digit of 7²³×8¹³=unit digit of 3×8=4

Question 2 What will be the unit digit of 888⁴⁴⁴+444⁸⁸⁸

2	4
5	6

A

Since the cyclicity of 8 is 4 and cyclicity of 4 is 2,

∴ 888⁴⁴⁴+444⁸⁸⁸=888^4×111+444^2×444

∴ last digit of 888⁴⁴⁴ and 444⁸⁸⁸=last digit of 8⁴ and 4² = 6 and 6

Thus unit digit =6+6=12 ∴ unit digit is 2

Question 3 Find the last digit of 3⁵⁷

3	6
1	7

A

3 has the cyclicity of 4. 3⁵⁷= 3^4×14+1

∴ last digit of 3⁵⁷=last digit of 3¹ = 3

Question 4 What will be the unit digit of 4567¹⁵⁷⁸

7	1
9	3

C

Since the cyclicity of 7 is 4, 4567¹⁵⁷⁸= 4567^4×394+2

∴ last digit of 4567¹⁵⁷⁸=last digit of 7² = 9

Question 5 What will be the unit digit of 199!

1	2
9	0

D

Factorial 199 = 1×2×3×4×5×..9×10×11..198×199 Thus the unit digit will be 0 (a number multiplied by 10 always end with 0)