Remainders

Over the past month or so, I have been trying to figure out the most efficient way to calculate questions like:

1. What will be the unit digit of 89⁷⁷ ,what is the tens digit of 89⁷ ,what will be the last three digit of 78⁶⁶ and so on.

2. What will be the remainder when 78⁶⁶ is divided by 5,7,9 etc.

3. What will be the remainder when 78⁶⁶⁶⁶⁶⁶ is divided by 5,17,24,35 etc.

Eventually, I find out a general methord to solve all these. Believe me its much simpler than it looks, although there are a few more ways to solve these problems (and I can share them with you too) but for now lets try to master one general methord.

Before that lets learn a small lesson to find relative primes of a number less than number itself [ Two relative prime numbers are those which have no common factor besides 1, such as 57 and 59], to find relative primes of number x we can use x(1-1/p₁)(1-1/p₃)(1-1/p₂)...................(1-1/p_n). Where p₁,p₂,p_n are prime factors of x.

So, relative primes of 9 are 9(1-1/3)=6 [as 3 is the only prime factor of 9 besides 9 and 1 which are not prime]. And those are 1,2,4,5,7,8.

Relative primes of 12 are 12(1-1/2)(1-1/3)=4 and those are 1,5,7,11.

Remember relative primes of prime number is always prime number -1. For eg. 5(1-1/5)=4.

So, lets take our first task

Unit digit of 89⁷⁷?

What we will do is try to divide it with 10, [ tell me why 10? or just remember that to find unit digit you have to divide by 10].

Now relative primes of 10 are 10(1-1/2)(1-1/5)=4 (remember it) [1,3,7,9]

Now what we do is divide the power(77) with number of relative primes(4).

So divide 77 by 4 you will get 19 remainder 1 [when you divide 89⁷⁶ by 10 you will get 0 as remainder] so you are left with 89¹ [one here is your remainder] divided by 10 and that is easy. So the remainder is 9 [89/10 gives 9 as remainder]. And 9 is your unit digit.

Easy isn't it?

Now lets try to find unit digit of 24⁹⁸. As we know 10 has 4 relative primes, so 98/4 [power/relative primes] will give 2 as remainder so we are left with 24² divided by 10, that is 576/10 gives remainder 6 and that is your unit digit.

Now try these

Find the unit digit of 89⁷⁸,23⁶⁷,25¹¹

To find tens digit or last two digits, we will follow the same above technique except for this time we will divide by 100. 100 has 40 relative primes [solved from methord told above]

So to find tens digit of 89⁸³ we will divide 83 by 40 [power by relative primes]. We will get 3 as remainder. So we are left with 89³ divided by 100. Do it manually by using Criss Cross multiplication or mental multiplication, we will get 69 as last two digits with 6 as required answer.

Similarly we can do it for last three digits, methord remains same but this time we will divide by 1000.

Now lets try this one “What will be the remainder when 78⁶⁶ is divided by 5”.

First find relative primes of 5 [ 5 is prime so its relative primes are 4]

Now divide the power of 78⁶⁶ by 4, so we get 2.

Now we are left with 78⁶⁶ divided by 5. Do it manually will give remainder as 4.

Now try these

What will be the remainder when 78⁶⁶ is divided by 5,7,9,11.

Suppose the base of numerator and denominator are not relative primes then take out the relative prime from Nr and Dr and multiply it in the end.

For example, to find the remainder when 78⁶⁷ is divided by 8.

So what we will do is take out 2 from base of Nr and Dr. Hence we are left with

2*{39⁶⁷/4}. Applying our usual methord to get remainder as 3 multiplying with 2 to get the result as 6.

To find remainder of something like 78^6567689divided by 15 (or any number), we apply our technique recursively.

Relative primes of 15 are 8. So we divide 65⁶⁷⁶⁹ by 8.

Relative primes of 8 are 4, so we divide 65⁶⁹ by 4

Relative primes of 4 are 2, so we divide 69 by 2 to get 1 as remainder.

Now we have 65⁶⁷¹ to divide by 4, so it will give remainder as 3.

Now we have 65³ to divide by 8, so it will give remainder as 1.

Now divide 78 by 15 to get remainder as 3.

Every subsequent year there is a question for working out unit digit/tens digit/remainder of a number in CAT. Every other competitive examination like FMS,IIFT surely has one or two such questions. CAT 08 had a question like “What will be the tens digit of 7²⁰⁰⁸?” Although this was a releatively easy question as 7 repeats its tens digit as well like

7¹=07

7²=49

7³=343

7⁴=2401

7⁵=16807

Well I hope you know the rest of the trick.

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If you want to know why this trick works write to me at friggere.Sumit@gmail.comuntil then bbye and all the best!!