Aptitude - Numbers - Discussion

The easiest way to do is;

1. Check digit at the unit place,
if it is 0, 1, 5, or 6, then we will have the same digit at unit place i.e unit digit for 0 is 0, 1 is 1, 5 is 5 and for 6 is 6.

If it is 2, 3, 4, 7, 8, or 9 then divide their power by 4 and find the remainder. Put the remainder as the power of the unit place of the given number.

In this example,
For (6374)^1793.


4 is at unit place of given no., so divide 1793 by 4 as mentioned above and we get the remainder i.e. .1

Now put the remainder as a power of 4 (which is unit place of given no.), therefore, 4^1= 4 this is required unit digit.

For (625)^317.
5 is at unit place, therefore for 5 it will be the same i.e. 5

For (341)^491.
1 is at unit place, therefore required no is same i.e 1.

Now multiply 4*5*1= 20 and final unit digit is 0.

Friends. There is a simple way to get this answer.

Consider the unit digit means the last digit. Here in this question, 4 is the unit digit of 6374 and 5 is the unit digit of 625 and 1 is the unit digit of 341.

We want to find the unit digit. So avoid the other numbers.

So, the next step is a simple method we only using the logic behind the multiplication.

(4)^any odd no = 4.
(4)^any even no = 6.
(5)^any no = 5.
(1) any no =1 what I meant that.

4^1 = 4.
4^2 = 16.
4^3 = 64.

i.e, the unit digit are 4 & 6 and this is repeating. Did you get my point?

So by using this method.

(4) ^1793 * (5) ^317 * (1) ^491 = ?
4 * 5 * 1 = 20.

So the unit digit is 0.

Hope, this help you. Thank you.

So here is the concept;

Cyclicity rule Table for numbers.

1 -1
2 - 4 (2^1 =2 ,2^2=4 ,2^3=8, 2^4=6) 2^5=2 ,2^6=4 .
Again the numbers repeat there are 4 distinct values.

2, 4, 6, 8 so (2 - 4) now the same goes for all other numbers.

3 - 4 (3^1=3, 3^2=9, 3^3=7, 3^4=1),
4 - 2 (4^1=4, 4^2=6 ,4^3=4, 4^4=6),
5 - 1 (5^1=5, 5^2=5, 5^3=5, 5^4=5),
6 - 1 (6^1=6, 6^2=6, 6^3=6, 6^4=6),
7 - 4 (7^1=7, 7^2=9, 7^3=3, 7^4=1),
8 - 4 (8^1=8, 8^2=4, 8^3=2, 8^4=6),
9 - 4 (9^1=9, 9^2=1, 9^3=9, 9^4=1).

[(6374)^1793 x (625)^317 x (341)^491].
(4)^1793 x (5)^317 X (1)^491.

1793 % 4= 1 so (4)^1 = 4,
317 % 5 = 2 so (5)^2 = 5,
491 % 1 = 0 so (1)^0 = 1.

Therefore 5 x 4 x 1 = 20 here unit digit is 0 => answer.

Look Friends its Simple:

Just take the last digit of number and last digit of power.

1). Here given is 6374^1793 means you should take 4^3 which is 64
means for 4^3 (4*4=16 and 16*4 is 64 or 6*4 is 24 whose unit digit is 4(remember for finding unit digit always take last digit after multiplication).

2). 625^317 means 5^7 here remember any power to 5 will always have 5 as its unit digit. So here the no is 5.

3). 341^491 means 1^1 is always 1 in its unit digit.

So from 1, 2, and 3 we got three numbers ie 4, 5, 1.

Now multiply these numbers and take the last digit of the product which will be your answer(4*5*1=20).

So 0 is your answer.

Hi guys,

These type of problems are solved by cyclicity rule.

Here this is the problem.

6374)^1793x(625)^317x(341)^491.

First term's unit digit is 4 cylicity for 4 is 4 (4^1) , 6 (4^2) , 4 (4^3) , 6 (4^4). This cycle continues in the same pattern. (Remember I only consider unit digits). So cyclicity is 2.

Now check first term is power 1793 is divisible by 2 or not. Clearly it shows that it will leave reminder 1. Which means unit digit is 4 for first term.

Second term unit digit 5, and last term is 1 so the answer is 4X5X1=20. Unit digit is 0. Note: Make a table for cylicity for numbers 1 to 9. Then it will become very handy.

Remember, all digits at the power of 5 gives the value having unit place containing the original digit. Special digits (0,1,5,6) raised to any power always give the value having unit place containing the original digit.

In this, 1793/5= ### and remainder is 3,
Unit value of 6374 is to be raised to power of 3 = 4^3= Unit value is 4.

Similarly 317/5= ### and remainder is 2,
Unit value of 625 is to be raised to power of 2 = 5^2= Unit value is 5.

Last factor comes in special digit i.e. 1 and will always give 1 at unit place,if raised to any power.

Hence, the unit place of the result will be 4x5x1= 20; 0 at unit place.

Follow this Rule.

Unit Digit of Base, Powers, Unit digit of product.

0, 1, 5, 6, any powers, same digit.

3, 7, 9, power exactly divisible by 4, 1.

2, 4, 8 " 6.

So in this problem: {(6374)^1793x(625)^317x(341)^491)}.

= (6374)^1793 unit digit is 4.

Apply the above rule: Divide 1793 by 4, we get remainder as 1.

Take the remainder to the power of unit digit's i.e. 4^1.

So we get (6374)^1793 = 4. For (625) ^317 same digit as the unit digit.

So we get (625)^317 = 5. For (341)^(491) same digit as the unit digit.

So we get (341)^(491) = 1.

Finally (4*5*1) = 20.

Thus unit digit is 0 = Answer.

Unit Digit of Base, Powers, Unit digit of the product.

0, 1, 5, 6, any powers, same digit.
3, 7, 9, power exactly divisible by 4, 1.
2, 4, 8 " 6.

So in this problem: {(6374)^1793 x (625)^317 x (341)^491)}.
= (6374)^1793 unit digit is 4.

Apply the above rule: Divide 1793 by 4, we get remainder as 1.

Take the remainder of the power of unit digit's i.e. 4^1.

So we get (6374)^1793 = 4. For (625) ^317 same digit as the unit digit.
So we get (625)^317 = 5. For (341)^(491) the same digit as the unit digit.
So we get (341)^(491) = 1.
Finally (4 * 5 * 1) = 20.

Thus unit digit is 0 = Answer.

The process generally is like this

Say 4^0=1 5^0=1 1^0=1 (1 time)
4^1=4 5^1=5 1^1=1 (2nd time)
4^2=16 5^2=25 1^2=1 (3rd time)
4^3=64 5^3=625 1^4=1 (4th time)

So as per problem units number must be the result of
(4^1793)*(5^317)*(1^491)

From the above calculations done it is evident that
1^anything =1 and 5^any positive number gives 5 as units digit
Similarly with 4 powers

Now take 4^1793.......First divide the power 1793/4 gives remainder 1.

So Units digit is (4^1)*5*1=20 whose
units digit is 0

(6374)^1793.

= Here unit digit is 4 then we can write (4)^1793.

As we know that multiple of 4 will be = 4, 16, 64, 256 = Here unit digit is 4 and 6 continuing coming then we have to take 2 nos. which should come 4 and 6 that's why we took.

= (4)^1793.

= 4*(4)^1792 (we can write it).

= 4*(4^2)^896.

Take only unit digit = 4*16 = 4*6 = 24 = 4.

Same as:

Unit digit in (625) 317 = Unit digit in (5) 317 = 5.

Unit digit in (341) 491 = Unit digit in (1) 491 = 1.

Answer:

Required digit = Unit digit in (4 x 5 x 1) = 0.