Remainder Theorem of Numbers
Remainder as a topic confuses the most. So we, www.gyaanshalaa.blogspot.com , are trying our best to impart you the deepest knowledge and ample of questions to give you better
understanding and clearing all your doubts.
Before going to the concept of Remainders, it is first required to have an idea about the basic
terminologies related to it like Dividend, Divisor, Quotient and Remainder.
KEY HIGHLIGHTS : Always remember that on dividing a number by N , the range of remainders is always from 0 to (N - 1) .
Remainder Theorem rule 1:
Remainder of the expression can be expressed as Positive remainder and Negative remainder.
But primarily, by definition remainders are always Non - Negative. Hence the final answer must be
positive only. Just for the sake of simplification, we calculate the remainder in negative. But it has to
be converted into positive while answering.
Method to convert Negative remainder into Positive remainder.
Add Divisor to Negative Remainder and obtain the Positive Remainder.
Subtract Positive Remainder from Divisor and you will get the Negative Remainder.
Example : Find the remainder when 51 is divided by 8 ?
-5
3
6
2
Sol :
+ 3 -5
↖ ↗
51
━━
8
In this question, Positive Remainder = +3
Negative Remainder = -5
But the remainder has to be positive ; Answer is 3.
Example : Find the remainder when 117 is divided by 15 ?
12
-3
7
8
Sol :
+ 12 -3
↖ ↗
117
━━
15
In this question, Positive Remainder = +12
Negative Remainder = -3
But the remainder has to be positive ; Answer is 12.
In the above question, finding the Negative Remainder is comparatively easier.
Remainder Theorem rule 2:
The remainder of the expression
{ A x B + C x D }
━━━━━━━━
N
will be same as the remainder of the
{ Aᵣ x Bᵣ + Cᵣ x Dᵣ }
Expression ━━━━━━━━
N
Where
Aᵣ = Aᵣ is the remainder when “A” is divided by “ N “.
Bᵣ = Bᵣ is the remainder when “B” is divided by “ N “.
Cᵣ = Cᵣ is the remainder when “C” is divided by “ N “.
Dᵣ = Dᵣ is the remainder when “D” is divided by “ N “.
Examples : Find the remainder when { 46 x 73 + 86 x 91 } is divided by 8.
2
6
8
0
Sol :
+6 or -2 +1 or -7 +6 or -2 +3 or -5
↖ ↑ ↑ ↗
{ 46 x 73 + 86 x 91 }
━━━━━━━━━━━
8
For the sake of less complicated calculation, you can take any type of remainders
PART 1 :
Case 1 : +6 x +1 = R (+6)
Case 2 : +6 x -7 = -42 = R(+6)
Case 3 : -2 x +1 = -2 = R(+6)
Case 4 : -2 x -7 = 14 = R(+6)
PART 2 :
Case 1 : +6 x +3 = +18 = R(+2)
Case 2 : +6 x -5 = -30 = R(+2)
Case 3 : -2 x +3 = -6 = R(+2)
Case 4 : -2 x -5 = +10 = R (+2)
6 + 2 8
━━━━ = ━━━ = R(0)
8 8
Remainder Theorem rule 3:
For Simplification , you should try to cancel out the part from the Numerator and Denominator as
much as possible then the final remainder must be multiplied with the canceled number so as to
get the correct answer.
Example : Find the remainder when 42 is divided by 4.
1
2
4
None of these ━━━━
Sol : 42 21
━━━━ = ━━━━ = R(1)
4 2
Final Remainder = R(1) x 2 = R(2)
Remainder Theorem rule 4:
Remainder of a number with power
There are two rules which are effective to deal with large powers.
If expression is in the form of (ax + 1)ⁿ , remainder will be 1 directly. No matter how
large the value of power “n”. ━━━━
a
If expression is in the form of (ax - 1)ⁿ , remainder will be (-1)ⁿ .
━━━━
a
If n = even number, Remainder will 1.
If n = odd number , remainder will be (-1).
Example : Find the remainder when 91⁶⁷ is divided by 5.
-1
1
2
None of these
Sol : 91⁶⁷ = ( 5 x 18 + 1) ⁶⁷ = R (1) ⁶⁷
━━━ ━━━━━━━
5 5
The remainder turns out to be (1)⁶⁷ = R(+1)
Example : Find the remainder when 87⁹⁷ divided by 8.
-1
1
7
-2
Sol :
87⁹⁷ = (8 x 11 - 1)⁹⁷ = R (-1)⁹⁷ [ HINT : ( - ) ^odd = - ]
━━━ ━━━━━━━
8 8
The remainder turns out to be (-1)⁹⁷= R(-1) = R(7)
Example : Find the remainder when 129⁹⁷² divided by 5.
2
1
-1
-2
Sol : 129⁹⁷² = ( 5 x 26 - 1)⁹⁷² = R (-1)⁹⁷² [ HINT : ( - ) ^even = + ]
━━━ ━━━━━━━
5 5
The remainder turns out to be (-1)⁹⁷²= R(+1)
KEY HIGHLIGHTS :
If a denominator is perfectly divisible by any one number of given numerator expression values
then the remainder of the whole expression is ZERO.
Example : Find the remainder when 124 x 175 x 769 x 856 is divided by 25.
12
6
9
0
Sol : 124 x 175 x 769 x 856
━━━━━━━━━━ = R(0)
25
Because 175 is completely divisible by 25. Hence turns the remainder into Zero.
PRACTICE MATERIAL :
Find the remainder when 123 + 456+ 789 + 921 + 457 is divided by 11.
7
8
-3
9
Sol : 123 + 456 + 790 + 921 + 479 (+2) + (+5) + (-2) + (-3) + (-5)
━━━━━━━━━━━━━ = ━━━━━━━━━━━━━
11 11
= R(-3)
Remainder = 11+(-3) = R(8)
Find the remainder when 123 x 432 x 739 x 463 is divided by 19.
4
6
9
7
Sol : 123 x 432 x 739 x 463 = (+9) x (-5) x (-2) x (+7)
━━━━━━━━━━ ━━━━━━━━━━
19 19
= (+9) x (+10) x (+7)
━━━━━━━━━
19
= (+9) x (-9) x (+7) = R (+7)
━━━━━━━━
19
The Remainder of the expression is 7.
Find the remainder when 2⁶⁶ is divided by 16.
4
7
0
15
Sol : 2⁶⁶ (2⁶⁴) x (2)² (16¹⁶) x (2)²
━━━ = ━━━━━ = ━━━━━ = R(0)
16 16 16
Find the remainder when 7⁶⁶ is divided by 14.
7
-7
1
0
SOl : Here 14 = 2 x 7. (Cancel 7 form the numerator and denominator)
7⁶⁵ (8-1) ⁶⁵ (-1) ⁶⁵
━━━ = ━━━ = ━━━ = R(-1) [ HINT : ( - ) ^odd = - ]
2 2 2
Multiplying remainder by the cancelled value : R(-1) x 7 = R(-7)
Hence the final answer is [14 +(-7)] = 7.
5. Find the remainder when 21⁸⁷⁵ is divided by 17.
4
9
13
6
Sol :
21⁸⁷⁵ (17 + 4)⁸⁷⁵ 4⁸⁷⁵ (4x4)⁴³⁷ x 4 16 ⁴³⁷ x 4
━━━ = ━━━━━ = ━━━ = ━━━━━━ = ━━━━━
17 17 17 17 17
(17-1)⁴³⁷ x 4 (-1)⁴³⁷ x 4
= ━━━━━━ = ━━━━━━ = R(-4)
The Final remainder is [17+ (-4)]= 13
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