Problem Solving

What is the digit in the units place in the expansion of (288)^81+(43)^231+(15)^67?
I don't have the answer options or the OA. Detailed explanations would be appreciated.

knight247 wrote:What is the digit in the units place in the expansion of (288)^81+(43)^231+(15)^67?
I don't have the answer options or the OA. Detailed explanations would be appreciated.

See this is a very easy question...
Now start checking from 1st number it is (288)^81
Find the last digit of powers of 8 so it is 8, 4, 2, 6, 8 .... so after 4 times 8 comes again
hence unit digit of 288^81 = 288^4x20+1 is 8.

Now start checking 2nd number it is (43)^231
Find the last digit of powers of 3 so it is 3, 9, 7, 1, 3 .... so after 4 times 3 comes again
hence unit digit of 43^231 = 288^4x57+3 is 7.


Now start checking 3rd number it is (15)^67
Find the last digit of powers of 5 so it is 5, 5, 5, 5, .... so after 1 times 5 comes again
hence unit digit of (15)^67 is 5.

So last digit of given expression = [spoiler]last digit of 8+ 7+ 5 = 0 answer[/spoiler]

hi,
(288)^81 +(43)^231 + (15)^67

for unit digit we can equate it to

(8)^81 + (3)^231 + (5)^67

now for,
2,3,7,8 ===> repeats unit digits at every 4 times

4 , 9 ===> repeats unit digits at every 2 times

0,1,6,5 ===> repeats unit digits at every 1 time


for 8 ==> 81/4 == reminder 1 (~8^1) ===> unit digit 8

for 3 ==> 231/4 == reminder 3(~3^3) ===> unit digit 7

for 5 ==> 67/2 == remainder 1 (~5^1) ===> unit digit 5


Hence unit digit = 8 + 7+ 5 =20

Ans= 0


whats the correct ans?

This is my first ans post... kindly advice if any mistake in my method.

br

ajay

knight247 wrote:What is the digit in the units place in the expansion of (288)^81+(43)^231+(15)^67?
I don't have the answer options or the OA. Detailed explanations would be appreciated.

(288)^81+(43)^231+(15)^67

8^81 + 3^231 + 5^67


8 has a power pattern of 4: 8,4,2,6 and then it repeats

81/4 = 20 r 1, so pattern repeats for 20 times and + 1 to end at 8, so unit digit here is 8


3^231 has a unit digit of 7

5's power pattern is only 5, so 5^67 has a unit digit if 5

8+7+5 = 20, so unit digit is 0