Greatest n-digit and Least n-digit Number Exactly Divisible by a Number
(a) To find out the greatest n-digit number exactly divisible by a divisor ‘d’, we use Method 1 (1.5.1) ∴ the required number = Greatest n-digit number − remainder. (b) To find out the least n-digit number exactly divisible by a divisor ‘d’, we use Method 2 (1.5.1), because if we use method 1, then subtracting any number from the n-digit least number will reduce it to (n − 1) digit number. ∴ the required number = Least n-digit number + (divisor − remainder) Example : Find the (a) greatest 3-digit number divisible by 13. (b) the least 3-digit number divisible by 13. Solution : (a) 13) 999 ( 11 ∴ the required 3-digit greatest number = 999 − 11 = 988 Solution :(b) 13) 100 ( 9 ∴ the required 3-digit least number = 100 + (13 − 9) = 104. Remainder Rules |
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Rule 1
This rule is applied to find the remainder for the smaller division, when the same number is divided by the two different divisors such that one divisor is a multiple of the other divisor and also the remainder for the greater divisor is know.
If the remainder for the greater divisor = r,
and the smaller divisor = d, then
Rule-1 states, the when r > d, the required remainder for the smaller divisor will be the remainder found out by dividing the ‘r’ by ‘d’. [Case I]
and when r < d, then the required remainder is ‘r’ it self. [Case II]
Example : If a number is divided by 527, the remainder is 42.
What will be the remainder if it is divided by 17?
Solution : Here the same number is divided by two divisors : 527 and 17.
Now,527 = 31, so, 527 is a multiple of 1717Hence Rule 1 can be applied.
Remainder for the greater divisor (i.e., for 527) = 42
Smaller divisor = 17.
So,
17) 42 ( 34 8 = required remainder for smaller divisor (i.e. 17)Hence, if 527 is divided by 17, the remainder will be 8.
Rule 2
If two different numbers a and b, on being divided by the same divisor leave remainders r1 and r2respectively, then their sum (a + b) if divided by same divisor will leave remainder R, given by
R = (r1 + r2) − divisor
⇒ The required remainder R = sum of remainders − divisor
(When sum is divided)
Note: If becomes negative in the above equation, then the required remainder will be the sum of the remainders.∴ the required remainder = sum of remainders
Example: Two different numbers, when divided by the same divisor, leave remainders 15 and 39 respectively, and when their sum was divided by the same divisor, the remainder was 7. What is the divisor?
Solution: Using the Rule 2
7 = (15 + 39) − divisor
⇒ divisor = 47
Example : Two different numbers, when divided by 47, leave remainders 13 and 23 respectively. If their sum is divided by the same number 47, what will be the remainder?
Solution : Using Rule 2,
The required remainder = (13 + 23) − 47
= −11
Since the remainder is (−) ve, so, the actual remainder will be 23 + 13 = 36 (refer to NOTE under Rule 2)
Rule 3
When two numbers, after being divided by the same divisor leave the same remainder, then the difference of those two numbers must be exactly divisible by the same divisor.
Example : Two number 147 and 225, after being divided by a 2-digit number, leave the same remainder. Find the divisor
Solution : By Rule 3, the difference of 225 and 147 must be perfectly divisible by the divisor.
The difference = 225 − 147 = 78
Now, 78 = 13 × 2 × 3.
Thus, 1-digit divisors = 2, 3 and 2 × 3
2-digit divisors = 13 × 2, 13 × 3, 13, 13 × 2 × 3
∴ the possible divisors are 26, 39, 13, 78.
Rule 4
If a given number is divided successively by the different factors of the divisor leaving remainders r1,r2 and r3 respectively, then the true remainder (i.e. remainder when the number is divided by the divisor) can be obtained by using the following formula :
True remainder = (first remainder) + (second remainder × first divisor)
+ (third remainder × first divisor × second divisor)
Example : A number, being successively divided by 5, 7 and 11 leaves 3, 1, 2 as remainders respectively. Find the remainder if the same number is divided by 385.
Solution : Here, the divisor is 385, whose factors are 5, 7 and 11.
∴ by Rule 3,
True remainder (i.e. remainder when divided by 385) = 3 + (1 × 5) + (2 × 5 × 7)
= 3 + 5 + 70
= 78
Rule 5
When (x + 1)n is divided by x, the remainder is always 1, where x and n are natural numbers.
Example : What will be the remainder when (17)21 is divided by 16?
Solution : (17)21 = (16 + 1)21,
∴ when (16 + 1)21 is divided by 16, the remainder = 1.
Rule 6
When (x − 1)n is divided by x, then
the remainder = 1, when n is an even natural number
but the remainder = x − 1, when n is an odd natural number.
Example : What will be the remainder when (29)75 is divided by 30?
Solution : (29)75 + (30 − 1)75, here index = 75 (which is odd) so, when (30 − 1)75 is divided by 30, the remainder will be x − 1 = 30 − 1 = 29.
This rule is applied to find the remainder for the smaller division, when the same number is divided by the two different divisors such that one divisor is a multiple of the other divisor and also the remainder for the greater divisor is know.
If the remainder for the greater divisor = r,
and the smaller divisor = d, then
Rule-1 states, the when r > d, the required remainder for the smaller divisor will be the remainder found out by dividing the ‘r’ by ‘d’. [Case I]
and when r < d, then the required remainder is ‘r’ it self. [Case II]
Example : If a number is divided by 527, the remainder is 42.
What will be the remainder if it is divided by 17?
Solution : Here the same number is divided by two divisors : 527 and 17.
Now,527 = 31, so, 527 is a multiple of 1717Hence Rule 1 can be applied.
Remainder for the greater divisor (i.e., for 527) = 42
Smaller divisor = 17.
So,
17) 42 ( 34 8 = required remainder for smaller divisor (i.e. 17)Hence, if 527 is divided by 17, the remainder will be 8.
Rule 2
If two different numbers a and b, on being divided by the same divisor leave remainders r1 and r2respectively, then their sum (a + b) if divided by same divisor will leave remainder R, given by
R = (r1 + r2) − divisor
⇒ The required remainder R = sum of remainders − divisor
(When sum is divided)
Note: If becomes negative in the above equation, then the required remainder will be the sum of the remainders.∴ the required remainder = sum of remainders
Example: Two different numbers, when divided by the same divisor, leave remainders 15 and 39 respectively, and when their sum was divided by the same divisor, the remainder was 7. What is the divisor?
Solution: Using the Rule 2
7 = (15 + 39) − divisor
⇒ divisor = 47
Example : Two different numbers, when divided by 47, leave remainders 13 and 23 respectively. If their sum is divided by the same number 47, what will be the remainder?
Solution : Using Rule 2,
The required remainder = (13 + 23) − 47
= −11
Since the remainder is (−) ve, so, the actual remainder will be 23 + 13 = 36 (refer to NOTE under Rule 2)
Rule 3
When two numbers, after being divided by the same divisor leave the same remainder, then the difference of those two numbers must be exactly divisible by the same divisor.
Example : Two number 147 and 225, after being divided by a 2-digit number, leave the same remainder. Find the divisor
Solution : By Rule 3, the difference of 225 and 147 must be perfectly divisible by the divisor.
The difference = 225 − 147 = 78
Now, 78 = 13 × 2 × 3.
Thus, 1-digit divisors = 2, 3 and 2 × 3
2-digit divisors = 13 × 2, 13 × 3, 13, 13 × 2 × 3
∴ the possible divisors are 26, 39, 13, 78.
Rule 4
If a given number is divided successively by the different factors of the divisor leaving remainders r1,r2 and r3 respectively, then the true remainder (i.e. remainder when the number is divided by the divisor) can be obtained by using the following formula :
True remainder = (first remainder) + (second remainder × first divisor)
+ (third remainder × first divisor × second divisor)
Example : A number, being successively divided by 5, 7 and 11 leaves 3, 1, 2 as remainders respectively. Find the remainder if the same number is divided by 385.
Solution : Here, the divisor is 385, whose factors are 5, 7 and 11.
∴ by Rule 3,
True remainder (i.e. remainder when divided by 385) = 3 + (1 × 5) + (2 × 5 × 7)
= 3 + 5 + 70
= 78
Rule 5
When (x + 1)n is divided by x, the remainder is always 1, where x and n are natural numbers.
Example : What will be the remainder when (17)21 is divided by 16?
Solution : (17)21 = (16 + 1)21,
∴ when (16 + 1)21 is divided by 16, the remainder = 1.
Rule 6
When (x − 1)n is divided by x, then
the remainder = 1, when n is an even natural number
but the remainder = x − 1, when n is an odd natural number.
Example : What will be the remainder when (29)75 is divided by 30?
Solution : (29)75 + (30 − 1)75, here index = 75 (which is odd) so, when (30 − 1)75 is divided by 30, the remainder will be x − 1 = 30 − 1 = 29.
While finding the unit digit a typical mind will first multiply all the numbers and will take the unit digit out of the final number.
Now, as we need to find out the unit digit, our focus must be on the unit digit only. In the same question, we just need to multiply the unit digits (leaving tens digit every time we get a double digit number) and we’ll get the answer. Let’s see. Steps to be taken: Multiplying the unit digits 567 x 693 x 391 x 453 x 188 We get, 7 x 3 = 21 (leaving the tens digit 2) 1 x 1 = 1 1 x 3 = 3 3 x 8 = 24 The answer will be 4. Now for the last few years unit digit questions are being asked in large powers of a number. Ex. Find the unit digit in 13137 The answer is 3. Before I explain this, kindly have a look at the patter how the powers of different digits behave. For n power of 0, we always get 0 as the unit digit. And same is the case with 1, 5 and 6. It means, 0n = 0, 1n = 1, 5n = 5, 6n = 6 For 0 as power of any number N we always get the unit digit as 1. It means, N0 = 1 Now, for 4 and 9 we have a similar patter that is as follows: For every odd power of 4 and 9 the unit digit is 4 and 9 respectively and for every even power of 4 and 9 the unit digit is 6 and 1 respectively. It means, 41 = 4 42 = 16 Similarly, for the power of 9 91 = 9 |
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92 = 81Now, different powers of 2 and 8 carry the same pattern. Let’s see what that is.
Unit digit in 21 = 2 & Unit digit in 81 = 8
Unit digit in 22 = 4 & Unit digit in 82 = 64 = 4
Unit digit in 23 = 8 & Unit digit in 83 = 512 = 2
Unit digit in 24 = 16 = 6 & Unit digit in 84 = 4096 = 6
Unit digit in 25 = 32 = 2 & Unit digit in 85 = 32768 = 8
Unit digit in 26 = 64 = 4 & Unit digit in 86 = 262144 = 4
Unit digit in 27 = 128 = 8 & Unit digit in 87 = 2097152 = 2
Unit digit in 28 = 256 = 6 & Unit digit in 88 = 16777216 = 6
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From the above table we can see that the “cycle” equals 4.
When the exponent is a multiple of 4, the unit digit is 6 in both the cases of base 2 or 8.
For example 24 has unit digit 6, 28 has unit digit 6.
Similarly, 84 has unit digit 6, 88 has unit digit 6.
For rest of the powers we will find the unit digit by calculating the remainder power
Ex: What is the unit digit in 13521354 ?
Solution:
As the unit digit in the base value is 2. We can re-write the number as 21354
To find the unit digit with base 2 we first need to convert the power in the multiples of 4 and for that we just need to divide the last two digits by 4.
(Divisibility Test of 4: To check the divisibility of a number by 4, the last two digits of the number must be either 00 or a multiple of 4.)
In the given power the last two digits are 54. When we divide 54 by 4 we get the remainder power as 2.
So our question shrinks to 22 that is very easy to find which is 4. The answer hence is 4.
Ex: What is unit digit in 178719 ?
Solution:
As the unit digit in the base value is 8. We can re-write the number as 8719
To find the unit digit with base 8 we first need to convert the power in the multiples of 4 as we did in the case of base 2.
In the given power the last two digits are 19. When we divide 19 by 4 we get the remainder power as 3.
The question hence shrinks to 83 the unit digit in which is 2. The answer, therefore, is 2.Now we are left with the last two digit which are 3 and 7. The two digits 3 and 7 too share a patterwhich is similar in itself.
Let’s have a look.
Unit digit in 31 = 3 & Unit digit in 71 = 7
Unit digit in 32 = 9 & Unit digit in 72 = 49 = 9
Unit digit in 33 = 27 = 7 & Unit digit in 73 = 343 = 3
Unit digit in 34 = 81 = 1 & Unit digit in 74 = 2401 = 1
Unit digit in 35 = 243 = 3 & Unit digit in 75 = 16807 = 7
Unit digit in 36 = 729 = 9 & Unit digit in 76 = 117649 = 9
Unit digit in 37 = 2187 = 7 & Unit digit in 77 = 823543 = 3
Unit digit in 38 = 6561 = 1 & Unit digit in 78 = 5764801 = 1
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From the above table we can see that the “cycle” equals 4.
When the exponent is a multiple of 4, the unit digit in both the cases is 1.
For rest of the powers we can use the remainder power method.
For example 34 has unit digit 1, 38 has unit digit 1.
Similarly, 74 has unit digit 1, 78 has unit digit 1.
For rest of the powers we can find the unit digit with the remainder powers after converting the maximum of powers into the multiples of 4.
Ex. Find the unit digit in 3193321
Solution:
Let’s first shorten the question by re-writing it as 3321
Now, dividing the last two digits of the power that is 21 by 4 we get the remainder 1.
(The powers which get converted into multiples of 4 will give unit digit 1. So, we neglect that.)
The question hence shrinks itself to 31 that is 3. The answer, therefore, is 3.Some Mixed Example:
Ex. 1. Find the unit digit in 3178 x 9613 x 7783
Solution:
Taking one base digit at a time, let’s first find out the unit digit in 3178
Using the method explained above, we get the remainder power 2 when we divide the last two digits of the power by 4.
So, the unit digit in 3178 = Unit digit in 378 = Unit digit in 32 = 9
Unit digit in 9613 = 9An odd power= 9
Unit digit in 7783 = Unit digit in 783 = Unit digit in 73 = Unit digit in 343 = 3
Now, we are left with three unit digits that go like:
9 x 9 x 3 = 3 (as we are concerned only with the unit digit)
The final unit digit is 3
Ex. 2. Find the unit digit in 1331677 − 77714
Solution:
Let’s first find the unit digit individually.
Unit digit in 1331677 = Unit digit in 31677 = Unit digit in 377 = Unit digit in 31 = 3
Unit digit in 77714 = Unit digit in 7714 = Unit digit in 714 = Unit digit in 72 = 9
The final equation we get is, 3 − 9
(Now, what you generally do when you subtract a bigger digit from a smaller one? You take a carry digit from the tens digit. Here, too we’ll do the same as when we are trying to find the unit digit, it doesn’t mean that the tens or hundreds digit doesn’t exit.)
The final unit digit hence will be = 3 − 9 = 13 − 9 = 4
The answer is 4.
Unit digit in 21 = 2 & Unit digit in 81 = 8
Unit digit in 22 = 4 & Unit digit in 82 = 64 = 4
Unit digit in 23 = 8 & Unit digit in 83 = 512 = 2
Unit digit in 24 = 16 = 6 & Unit digit in 84 = 4096 = 6
Unit digit in 25 = 32 = 2 & Unit digit in 85 = 32768 = 8
Unit digit in 26 = 64 = 4 & Unit digit in 86 = 262144 = 4
Unit digit in 27 = 128 = 8 & Unit digit in 87 = 2097152 = 2
Unit digit in 28 = 256 = 6 & Unit digit in 88 = 16777216 = 6
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From the above table we can see that the “cycle” equals 4.
When the exponent is a multiple of 4, the unit digit is 6 in both the cases of base 2 or 8.
For example 24 has unit digit 6, 28 has unit digit 6.
Similarly, 84 has unit digit 6, 88 has unit digit 6.
For rest of the powers we will find the unit digit by calculating the remainder power
Ex: What is the unit digit in 13521354 ?
Solution:
As the unit digit in the base value is 2. We can re-write the number as 21354
To find the unit digit with base 2 we first need to convert the power in the multiples of 4 and for that we just need to divide the last two digits by 4.
(Divisibility Test of 4: To check the divisibility of a number by 4, the last two digits of the number must be either 00 or a multiple of 4.)
In the given power the last two digits are 54. When we divide 54 by 4 we get the remainder power as 2.
So our question shrinks to 22 that is very easy to find which is 4. The answer hence is 4.
Ex: What is unit digit in 178719 ?
Solution:
As the unit digit in the base value is 8. We can re-write the number as 8719
To find the unit digit with base 8 we first need to convert the power in the multiples of 4 as we did in the case of base 2.
In the given power the last two digits are 19. When we divide 19 by 4 we get the remainder power as 3.
The question hence shrinks to 83 the unit digit in which is 2. The answer, therefore, is 2.Now we are left with the last two digit which are 3 and 7. The two digits 3 and 7 too share a patterwhich is similar in itself.
Let’s have a look.
Unit digit in 31 = 3 & Unit digit in 71 = 7
Unit digit in 32 = 9 & Unit digit in 72 = 49 = 9
Unit digit in 33 = 27 = 7 & Unit digit in 73 = 343 = 3
Unit digit in 34 = 81 = 1 & Unit digit in 74 = 2401 = 1
Unit digit in 35 = 243 = 3 & Unit digit in 75 = 16807 = 7
Unit digit in 36 = 729 = 9 & Unit digit in 76 = 117649 = 9
Unit digit in 37 = 2187 = 7 & Unit digit in 77 = 823543 = 3
Unit digit in 38 = 6561 = 1 & Unit digit in 78 = 5764801 = 1
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From the above table we can see that the “cycle” equals 4.
When the exponent is a multiple of 4, the unit digit in both the cases is 1.
For rest of the powers we can use the remainder power method.
For example 34 has unit digit 1, 38 has unit digit 1.
Similarly, 74 has unit digit 1, 78 has unit digit 1.
For rest of the powers we can find the unit digit with the remainder powers after converting the maximum of powers into the multiples of 4.
Ex. Find the unit digit in 3193321
Solution:
Let’s first shorten the question by re-writing it as 3321
Now, dividing the last two digits of the power that is 21 by 4 we get the remainder 1.
(The powers which get converted into multiples of 4 will give unit digit 1. So, we neglect that.)
The question hence shrinks itself to 31 that is 3. The answer, therefore, is 3.Some Mixed Example:
Ex. 1. Find the unit digit in 3178 x 9613 x 7783
Solution:
Taking one base digit at a time, let’s first find out the unit digit in 3178
Using the method explained above, we get the remainder power 2 when we divide the last two digits of the power by 4.
So, the unit digit in 3178 = Unit digit in 378 = Unit digit in 32 = 9
Unit digit in 9613 = 9An odd power= 9
Unit digit in 7783 = Unit digit in 783 = Unit digit in 73 = Unit digit in 343 = 3
Now, we are left with three unit digits that go like:
9 x 9 x 3 = 3 (as we are concerned only with the unit digit)
The final unit digit is 3
Ex. 2. Find the unit digit in 1331677 − 77714
Solution:
Let’s first find the unit digit individually.
Unit digit in 1331677 = Unit digit in 31677 = Unit digit in 377 = Unit digit in 31 = 3
Unit digit in 77714 = Unit digit in 7714 = Unit digit in 714 = Unit digit in 72 = 9
The final equation we get is, 3 − 9
(Now, what you generally do when you subtract a bigger digit from a smaller one? You take a carry digit from the tens digit. Here, too we’ll do the same as when we are trying to find the unit digit, it doesn’t mean that the tens or hundreds digit doesn’t exit.)
The final unit digit hence will be = 3 − 9 = 13 − 9 = 4
The answer is 4.