Aptitude - Numbers - Discussion

To find unit digit first we should know the periodic value
ex:-for 2

2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64 and so on.

Clearly observe that unit digit follows a cyclic form i.e after 4 terms it again repeating. So the period of 2 is 4.

Similarly for 3, 7, 8 also same period 4.

For 0, 1, 5, 6 period is always same as itself irrespective of their powers.

So to find a unit digit find the period of that number then divided the power value by the period.

ex: 3^65 period for 3 is 4.

Divide 65 by 4 we get reminder 1.
Now unit digit is 3^1 = 3.
6^59 =6.
7^71 =period for 7 is 4.

Divide 71 by 4 we get remainder 3
7^3 =343 => 3(consider unit digit).
So answer is 3*6*3 = 54 => 4 (unit digit is 4).

3^65 * 6^59 * 7^71.

= ? * 6 * ? // If the unit digit of base is [0,1,5,6] then answer will be same
= 3^1 * 6 * 7^3 // If the unit digit of base is other than [0,1,5,6] then divide the power with 4 and remainders will be one of these [0,1,2,3], take the remainder as the power of unit digit of the base.
=3 * 6 * 3 // unit digit of 7^3 will be [7*7*7 , take 49*7 and calculate only unit digit which is 9*7 = 63 and unit digit is 3]

=4 // [18 * 3 here calculate only unit digit which is 8*3 = 24 , and answer will be 4]

Unit digit in 34 = 1 unit digit in (34)16 = 1.

Unit digit in 365 = unit digit in [(34) 16 x 3 ] = (1 x 3) = 3.

Unit digit in 659 = 6.

Unit digit in 74 unit digit in (74) 17 is 1.

Unit digit in 771 = unit digit in [(74) 17 x 73] = (1 x 3) = 3.

Required digit = unit digit in (3 x 6 x 3) = 4.

How the unit digits are find? Please explain clearly.

Unit digit of 7^71wxplantion:

7^7=49 in dis unit digit is 9
Then 7 multiplied by 49 leaves unit digit as 3,
Then 7 multiplied again gives 1,
Again 7 multiplied gives 7.

Repaet d order like : 9,3,1,7,9,3,1,7,9,3,1,7
In the order, 11th place gives 1 which would be identical to 7^71.
Ex: lf it's 7^ 72 then the unit place would be 7.

Because this is shortcut for cyclic these types of some e.g.

1, 5, 6 = 0.

4, 9 = 2.

2, 3, 7, 8 = 4.

Rule 1. When remainder is 0 and no. At unit place is even no. Answer is always 6.

Rule 2. When remainder is 0 and no. At unit place is odd and is always 1.

@All.

According to me, the solution is;

3^65 * 6^59 * 7^71=?
The cycle city of 3 is 4=1
The cycle city of 6 is 1=6
The cycle city of 7 is 4=3
1* 6 * 3,
6 * 3 = 18 = 8/2 = 4.

E.g. 3^200 * 4^500?

Solution :-
3^200 * 4^500.
200/4 = 50, remainder is 0 , 3^4.
500/4 = 125, remainder is 0, 4^4.
= 3^4 * 4^4.
= 81 * 256.
= 1 * 6.
= 6 is the answer.

I don't know how to calculate the unit digit can you explain?

How you will say 3^4=1? I'm get confused can you explain me?

I did not get the answer how can I calculate the unit number please explain me anyone.

What is the unit digit in the product (4)^{66} x (6)^{59} x (5)^{71} ?