Aptitude - Numbers - Discussion

No dude not always by 4.

4 ka first power = 4 (unit digit 4).
4 ka square = 16 ( unit digit 6).
4 ka cube = 64. ( unit digit 4).
4 ka fourth power = 2401 ( unit digit 1).

4 ka fifth power = now check unit digits later on going to be repeat again. That is 4,6,4,1.

That's why. Unit digit restriction. Is. Upto 4th power in this case

That's why.. Divide the no by 754 by 4.. you will get 2 as a remainder.

By rule ,
Dividend = divisor *quotient +remainder.

Assume quotient = n.

754=4n+2.
n=188.

4137 power 754 = 7 ^ 4 ^188 + 7^2.
=9.

Just devide the power with 4, we get the remainder 2 and if we observe the last digits in the first four consecutive powers of 7, we get 7,9,3,1, and after that last digits will repeat again, that is every period of 4, the last digits will be repeated. So, as we got the remainder 2 when we divide the power by 4 the last digit would be 9. There is no need to expand 4173^2.

The units will be like.

7^1=7.
7^2=49; unit dig=9,

For 7^3 = 343; uit dig=3.
For 7^4=___1 (unit dig =1).

Then it follows as. 7, 9, 3, 1 consecutively. Here we have cycles of 4.

Therefore, 752 can be divided from 4.
752 = 1.
753 = 7.
754 = 9; by using the cycle the answer is 9.

7^754 = ((7^4)^187)*7^6 | 7^4=81 here unit digit of 81 is 1)
= ((1)^187)*(7^4)*7^2.| 7^4=81 here unit digit of 81 is 1)
= 1*1*49.
= 1*49.
= 1*9(in 49, the number 9 is unit number)
= 9.
So is divisible by 9 so the answer is 9.

4137^754,

Step 1: The unit digit of the exponent 4137 is 7,
Step 2: Divide 754 by 4 (754÷4); remainder = 2,
Step 3: From step 1 &2=> 7^2 =49.

So, the unit digit of 49 is 9.

Therefore, the answer is 9.

According to the cyclicity:

754/4 = remainder is 2.
In 4137^754 consider unit digit i.e 7^754.
replace the power with remainder i.e 7^2.
49 they asked unit digit so consider a unit digit.
Answer = 9.

Power = 754.
Divide by 4(7,9,3,1).
754/4 = 2(remainder).

Given number 4137 = 7(last digit).
Last digit^remainder.
7^2 = 49.

Last digit of 49 is 9(answer).

754 => 4*n + 2, Here n will be 188.

So, unit digit will be the unit digit of 7^2, ie 9.

If we consider (4137^2) ^377 is it right or is there compulsory to consider 4 as the power?

Is it always that the powers are divided by 4? If yes, kindly explain the concept.