GATE2017 CE-1: GA-8

Question

The last digit of $(2171)^{7}+(2172)^{9}+(2173)^{11}+(2174)^{13}$ is

• A. $2$
• B. $4$
• C. $6$
• D. $8$

Answer 1

We know that

For unit digit $1$,

• ${1}^{n}=1$ where $n$ is whole number

For unit digit $2,$

• $2^{1}=2$
• $2^{2}=4$
• $2^{3}=8$
• $2^{4}=16$
• $2^{5}=3{\color{Red}{2}}$
• Cyclicity of $2$ is $4.$

For unit digit $3,$

• $3^{1}=3$
• $3^{2}=9$
• $3^{3}=27$
• $3^{4}=81$
• $3^{5}=24{\color{Magenta}{3}}$
• Cyclicity of $3$ is $4.$

For unit digit $4,$

• $4^{1}=4$
• $4^{2}=16$
• $4^{3}=64$
• $4^{4}=256$
• $4^{5}=102{\color{Blue}{4}}$
• Cyclicity of $4$ is $2.$

For unit digit $6,$

• $6^{1}=6$
• $6^{2}=36$
• We can write last digit of $6$ is $6^{n}=6$ where $n\in N$

$(2171)^{7}+(2172)^{9}+(2173)^{11}+(2174)^{13}$

We want to find last digit, so we can focus only last digit.

$\implies(1)^{7}+(2)^{9}+(3)^{11}+(4)^{13}$

$\implies 1+(2^{4})^{2}\cdot 2^{1}+(3^{4})^{2}\cdot 3^{3}+(4)^{odd}$

$\implies 1+(6)^{2}\cdot 2^{1}+(1)^{2}\cdot 3^{3}+4$

$\implies 1+6\cdot 2+ 7+4$

$\implies 1+2+ 7+4$

$\implies 14$

So, last digit is $4$

The answer $(B)$ is the correct choice

Comments

Unit digit of $4:$

• $4^{1} = 4$
• $4^{2} = 16$

We can say that unit digit of $4$ is  $4^{\text{odd}} = 4$ and $4^{\text{even}} = 6$
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Unit digit of $9:$

• $9^{1} = 9$
• $9^{2} = 81$

We can say that unit digit of $9$ is  $9^{\text{odd}} = 9$ and $9^{\text{even}} = 1$

Answer 2

2171 (last digit 1) -> 1^n=1

2172 (last digit 2) -> 2 repeats its last digits as : 2,4,8,6,2..... => 9=4+4+1 => last digit = 2

2173 (last digit 3) -> 3 repeats its last digits as : 3,9,7,1,3..... => 11=4+4+3 => last digit = 7

2174 (last digit 4) -> 4 repeats its last digits as : 4,6,4..... => 13=2*6 + 1 => last digit = 4

1 + 2 + 7 + 4 =14

Answer : b

Answer 3

Divide all the powers by 4
2171^7=If you divide 7÷4 you get remainder 3. Now take units digit of 2171 i.e 1. Now take the remainder 3 as power of 1 i.e 1^3=3.
Now take 2172^9 divide 9 by 4 you get remainder as 1 then 2^1
So if you divide all the powers by 4 you get like this 1^3+2^1+3^3+4^1=34 we need units place so answer is 4
Option (b) is correct