The concept of finding the unit digit is important if you are preparing Number System for CAT. It is completely different from the concept of finding the last two digits. Let us take an example. What is the unit digit of the product 45 × 44? Well, if we multiply 45 and 44, the product is 1980, so the unit digit is 0. Now, what will be the answer if we take the numbers as 445 × 444. The product in this case is 197580. The unit digit is again 0. Now the question is, do we really need to multiply the given numbers to get the unit digit? The answer is no.

You can see that the 0 in the unit place is nothing but the product of 4 and 5 which are the unit places of the given numbers. Therefore, when we are finding the unit digit of the product or addition of the numbers, we just need their unit digits and not the whole number.

Let us take another example:

**Ex. **What is the unit digit of the product 124 × 428 × 322?

**Sol: **Here the given product is 124 × 428 × 322. The unit digits of the numbers here are 4, 8 and 2. Therefore, the unit digit of the product of these numbers is same as the unit digit of the product 4 × 8 × 2 which is 64 or 4. Hence, the unit digit of the product 124 × 428 × 322 is 4.

Now, while preparing the number system for CAT exam, the questions that might be asked are of the type where the unit digit of the numbers of the form a^{n} is asked. Before, we move further let us discuss some basics of the unit digits when the numbers are of the form a^{n}.

If a number has 0, 1, 5 or 6 at the unit place then any power of such number again has 0, 1, 5 or 6 at the unit place. It means that 15^{20} will have 5 at the unit place and 176^{125} will have 6 at the unit place.

Now let us take the powers of 2.

2^{1} = 2, 2^{2} = 4, 2^{3} = 8, 2^{4} = 16, 2^{5} = 32, 2^{6} = 64…

Now, after 2^{4}, the unit digit of the numbers is repeating. Therefore, the cycle of the unit digit of 2 is 4.

The cycle for the powers of 3 is 3^{1} = 3, 3^{2} = 9, 3^{3} = 27, 3^{4} = 81. After this, we have 3^{5} = 243 has unit digit 3 which is same as the unit digit of 3^{1}. Therefore, the cycle in this case is 4.

The cycle of 4 is of 2 steps, i.e., 4^{1} = 4 and 4^{2} = 16 as 4^{3} = 64 again has unit digit 4.

So, in case of 4, we can say that if the power of 4 is odd, then the unit digit will be 4 and if the power is even, then the unit digit will be 6.

The same is true for 9 also. The cycle of 9 is also of 2 steps, i.e., 9^{1} = 9 and 9^{2} = 81 as 9^{3} = 729 has unit digit 9 again.

So, in case of 9, we can say that if the power of 9 is odd, then the unit digit will be 9 and if the power is even, then the unit digit will be 1.

The cycle for the unit digit for the numbers 7 and 8 is also of 4 steps.

For 7, we have 7^{1} = 7, 7^{2} = 49, 7^{3} = 343, 7^{4} = 2401 as the steps of cycle. After the 4^{th} step, there will be repetition of the last digit.

For 8, we have 8^{1} = 8, 8^{2} = 64, 8^{3} = 512, 8^{4} = 4096. After the 4^{th} step, there will be repetition of the last digit.

We have seen above that the cycle for the unit digit of powers of 2, 3, 7 and 8 is of 4 steps whereas for the numbers 4 and 9, it is of two steps. Now, how to apply this in questions? For that, let us take an example.

Let us try to find the unit digit of 2^{17}. For this remember that the cycle for unit digit of 2 was of 4 steps and the steps were 2^{1} = 2, 2^{2} = 4, 2^{3} = 8, 2^{4} = 16. Now, in this question, the power of 2 is 17. Divide this power 17 by 4 (the number of steps in the cycle) and find the remainder as 1. Now this remainder tells us the step number of the cycle. So as in this case we are getting the remainder as 1, so it means that unit digit of 2^{17} will be same as in the first step of cycle of 2. The unit digit in the first step of cycle of 2 is 2. Therefore, the unit digit of 2^{17} is 2.

Let us now find the unit digit of 2^{255}. Here the power is 255 and the cyclicity of 2 is 4. Divide 255 by 4, the remainder is 3. This remainder 3, gives the step number of the cycle. Now in the third step of the cycle of 2, the unit digit is 8, therefore, the unit digit of 2^{255} is 8.

Let us find the unit digit of 3^{772}. Now, the cycle of 3 is again of the four steps and the steps here are 3^{1} = 3, 3^{2} = 9, 3^{3} = 27, 3^{4} = 81.

Divide the power 772 by 4 (the number of steps in the cycle) and find the remainder as 0. Now this remainder 0 means that the cycle of 3 is just ended, i.e., we are at the last step of the cycle and the unit digit in the last step of the cycle of 3 is 1. Therefore, the unit digit of 3^{772} is 1.

Let us now practice the following questions:

**1.** What is the unit digit of 32^{277} × 58^{145}?

**Sol: **Here, we have 32^{277} × 58^{145}.

The unit digit will be same as of 2^{277} × 8^{145} as for the unit digit we do not need the whole number but only the unit digit of the number.

The cyclicity of both 2 and 8 is 4. Divide the power of 2 i.e., 277 by 4 and get the remainder 1. Therefore, the unit digit of 2^{277} is the first step of the cycle of 2 i.e., 2.

Now divide the power of 8 i.e., 145 by 4 and get the remainder as 1 which again means that the unit digit will be the first step of cycle of 8 i.e., 8.

So, the final answer is 32^{277} × 58^{145} = 2 × 8 = 16 or 6 is the answer.

**2.** What is the unit digit of 256^{256} × 544^{544} + 123^{123}?

**Sol:** The unit digit of 256^{256} × 544^{544} + 123^{123} is same as of 6^{256} × 4^{544} + 3^{123}.

Now any power of 6 will have 6 at the unit place again. So, 6^{256} will have 6 at the unit place.

In 4^{544} the power is even, so its unit digit is 6.

Now divide the power of 3 i.e., 123 by 4 (the cyclicity of 3) and get the remainder as 3 which means that the unit place will be same as the 3^{rd} step of cycle of 3, which is 7.

Therefore, the final answer will be 6 × 6 + 7 = 36 + 7 = 43. So, the final answer is 3.

**3.** What is the unit digit of 33^{455} + 85^{747} – 28^{282}?

**Sol:** Here the unit digit of the 33^{455} + 85^{747} – 28^{282} will be same as that of 3^{455} + 5^{747} – 8^{282}.

Now 455 divided by 4 will leave remainder 3, so, the unit digit of 3^{455} will be 7. Also, any power of 5 will have 5 at the unit’s place. Therefore, the unit’s place of 5^{747} is 5.

Divided 282 by 4 and get the remainder as 2. So, the unit’s digit will be the second step of the cycle of 2 i.e., 4.

Therefore, the unit digit of 33^{455} + 85^{747} – 28^{282} = 7 + 5 – 4 = 8.

**4.** What is the unit digit of the number 33^{100!}?

**Sol:** The number here is 33^{100!}. The power 100! is a multiple of 4, so will give remainder 0 when divided by 4. Now the remainder 0 means that we are at the last step of the cycle of 3 and the unit digit of the last step is 1.

**5.** Find the unit digit of the number 632^{254} × 978^{546} – 129^{247}.

**Sol:** Here the unit digit of 632^{254} × 978^{546} – 129^{547} will be same as that of 2^{254} × 8^{546} – 9^{247}.

254 divided by 4 leaves the remainder 2. So, the unit’s digit of 2^{254} is 4.

546 divided by 4 leaves the remainder 2, So, the unit’s digit of 8^{546} is 4.

The power of 9 is 247 which is an odd number, therefore, the unit digit of 9^{547} is 9.

Therefore, the unit digit of 632^{254} × 978^{546} – 129^{547} = 4 × 4 – 9 = 7.

While preparing the number system for CAT examination, the concepts that are used to calculate the unit’s digit of a number are same as discussed above.

In the next article we will discuss about the last two digits of a number…

Till then Happy Learning!!!!!

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