what is the unit digit in 2^99
Answers
Answer:
What is the unit digit of 2^100?
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2^1 =2
2^2= 4
2^3=8
2^4=16
2^5= 32
2^6=64
Here we here we observe that after every 4 numbers, the units digit of 5th number gets repeated. That is, After 8 the the units digit of 5th number is 2 again.
So we conclude that, after 40 numbers the value will be repeated will be 6 similarly after 92 the value repeated will be 6 and so the units digit of 2^100= 6.
Alternatively,
2^100= (2^4) ^25 = (16)^25
Now, any power of 16 will yield units digit = 6.
Hence the units digit of 2^100= 6.
The unit digit in 2^99 is 8.
Given: 2^99
To Find: The unit digit in 2^99
Solution:
- It is to be noted that every digit (0,1,2, 3,…..9) when raised to power follows a pattern, i.e., there is a cyclicity in the values of units digits when raised to powers.
- Since we are concerned with the unit digit of 2 in this question, so we need to find the pattern that the unit's place of 2 follows:
2^1 - 2 [ unit digit is 2 ]
2^2 - 4 [ unit digit is 4 ]
2^3 - 8 [ unit digit is 8 ]
2^4 - 16 [ unit digit is 6 ]
2^5 - 32 [ unit digit is 2 ]
2^6 - 64 [ unit digit is 4 ]
2^7 - 128 [ unit digit is 8 ]
2^8 - 256 [ unit digit is 6 ]
2^9 - 512 [ unit digit is 2 ]
- So, if we look at it the unit’s digit pattern for 2 raised to a positive integer follows a pattern of 2, 4, 8, and 6 in that order.
- Since, the pattern repeats itself after every 4 times, so, to find the unit’s digit when 2 is raised to any positive integer power, take the power and divide it by 4.
- If the remainder is,
1 – it ends with 2
2 – it ends with 4
3 – it ends with 8
0(divisible by 4) – it ends with 6
We are given to find the unit digit in 2^99.
So, we shall divide 99 by 4 to check the remainder.
When 99 is divided by 4, we get a remainder of 3.
So, the unit digit must be 8.
Hence, the unit digit in 2^99 is 8.
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