Question

Let L = 1202^2 + 2^2021 Determine the unit digit of the number L^2021 .

Answer 1

Concept Introduction:-

A digit is among the components of a set that makes up a numerical system.

Given Information:-

We have been given that $L = 1202^2 + 2^{2021}$

To Find:-

We have to find that the unit digit of the number $L^{2021}$

Solution:-

According to the problem

If you think about just the units digit of $1202^2$ you can see the only things that contribute are the product of the units digits, which would give you $2 * 2 = 4$

Therefore the units digit of $1202^2$ is $4$

Similarly, Now think about $2^{2021}$

Let's just keep doubling numbers, but let's ignore any digit that isn't the units digit, since they can't affect the next unit digit anyway:

$2 -- > 4 -- > 8 -- > 6 -- > 2 -- > 4$

Here you can see the units are repeating. In fact, this sequence has a period of $4$, so every time you take $2n$ where n is a multiple of $4$, you get a $6$ in the unit digit.

If n has remainder $1$ on division by $4$, you get a $2$ in the units digit. If the remainder is $2$ you get $4$, and if the remainder is $3$ you get $8$.

So now we know that to find the unit digit of $2^{2021}$ we just need to know what remainder $2021$ leaves on division by $4$.

$2021$ leaves a remainder of $1$ on division by $4$, so we know the units digit of $2^{2021}$ is $2$.

Therefore the units digit of $1202^2 + 2^{2021}$ is $4 + 2 = 6$

Finally, we can do the same thing and think about the units digit of $L^{2021}$

If we keep multiplying by 6 and looking at the units digit, we get the sequence:

$6 -- > 6 -- > 6 -- > 6 -- >$

Therefore we can see that we'll always get a $6$. In which case the unit digit of $L^{2021}$ could only be a $6$.

Final Answer:-

The unit digit of the number $L^{2021}$ is $6$.

#SPJ3

Answer 2

Answer:

Concept:

The units digit is the digit that represents the integer's 'number of ones.' It is the number's rightmost digit, preceding any decimal point. A units digit can usually be found by glancing at the number and finding the rightmost number before the decimal. However, this is not always the case. In exponentiated numbers, the units digit must be determined. For example, rather than calculating the exact value and then finding the units digit, it is preferable to use an indirect method to determine the units digit of 260.

Given:

Let $L =$ $1202^2$ $+$ $2^{2021}$

Find:

Determine the unit digit of the number $L^{2021}$

Explanation with example:

The given $L=1202^2+2^{2021}$

Finding the remainder when a number is divided by 10 is one method of determining the units digit of a power. Another common and simple method

for determining the units digit of a number in the form

=> Determine the units digit in base '$x$' and call it '$l$'. ($x=24$, for example, the units digit in $24$ is $4$, Hence $l=4$)

=> Multiply the exponent '$y$' by $4$.

=> If the exponent '$y$' is exactly divisible by $4$, that is, when divided by $4$,   $y$' leaves a $0$ residual.

Then,

the units digit of $x^y$ is $6$, if $l=2,4,6,8$

the units digit of $x^y$ is $1$, if $l=3,7,9$

* When divided by $4$, '$y$' leaves a non-zero leftover '$r$'.

Then,

the units digit of $x^y=l^r$.

Answer:

According to the given explanation

the unit digit of $1202^2=4$

=> unit digit of $2^{2021}$

now, divide $2021$ with $4$

$2021$ leaves a remainder of $1$ while dividing with $4$

so, the units digit of $2^{2021}$ is $2$

∴ the units digit of $= 1202^2+2^{2021}$

                               $=4+2$

                               $=6$

=> similarly $L^{2021}$

$L=6$

$6*2=12$

so, the unit digit is $6$

therefore the unit digit of a number $L^{2021}$ is $6$

∴ the unit digit of the number $L^{2021}$ is $6$

#SPJ3