Remainder Theorem:
i.e. Sum of numerical values (ignoring sign) of positive remainder and negative remainder is equal to value of divisor.
Ex:- A number when divided by 13 gives – 4 as negative remainder. What will be the actual or positive remainder.
Sol:- actual remainder = 13 – 4 = +9
Ex:- A number when divided by 13 gives +4 as positive remainder. What will be the negative remainder.
Sol:- Negative remainder numerical value = 13 – 4 = 9
∴ Negative remainder = – 9
⟶ negative remainder is used as a concept to make calcultation easier otherwise positive remainder is the valid remainder
Ex:- Find remiander of given expresseion:
● Remainer of any number is less than the divisor.
Ex:- Remaninder of 5 can be 0, 1, 2, 3, or 4 not 5 or more than 5
Remainder of 7 can be 0, 1, 2, 3, 4, 5, or 6 not 7 or more than 7.
● Divide a number by 10 and find its remainder which will give its unit digit (last digit)
● Divide a number by 100 and find its remainder which will give its last two digits.
If remainder is single digit add ‘0’ before it.
● Given a fraction \(\frac{a}{b}\), if we simplify it by a factor of k i.e. \(\frac{{\frac{a}{k}}}{{\frac{b}{k}}} = \frac{c}{d}\) and remainder of \(\frac{c}{d}\) comes out to be R then remainder of \(\frac{a}{b}\) will be k.R
(Q). Find the remainder of \(\frac{{596}}{{15}}\) .
Sol:- By quick observation we find that 600 is divisible by 15 and 596 is near to 600.
∴ remainder = -4 = 15-4 = 11 Answer
(Q). Find the remainder of \(\frac{{674}}{{68}}\)
Sol:- By quick observation we find that 680 is divisible by 68 and is also near to 674
∴ remainder = -6 = 68 – 6 = 62 Answer
(Q). Find the remainder of
\(\frac{{14979 \times 455 \times 53 \times 72 \times 29}}{{100}}\)
⟶ Division by 2 ⟹ If Uₔ is even then 0
If Uₔ is odd then 1
⟶ Division by 3 ⟹ discard all digits which are multiple of 3 or their sum is multiple of 3 and take the remainder of sum of remaining digits.
Ex:- Find the remainder of \(\frac{{87005309476}}{3}\)
Sol:-
Ex:- Find the remainder of \(\frac{{9876545321}}{3}\)
Sol:-
⟶ Division by 4 ⟹ Take remainder of last two digits
Ex:- \(\frac{{98370568973}}{4}\)
Remainder = Remainder of \(\frac{{73}}{4}\) = 1 Answer
⟶ Division by 5 ⟹ If last digit is 0 or 5 then remainder is 0 else divide last digit by 5 and take remainder
Ex:- (\frac{{98370568973}}{5}\)
Sol:- Uₔ = 3 ∴ Remainder = \(\frac{3}{5}\) = 3 Answer
⟶ Division by 6 ⟹ Multiply the sum of all digits except unit digit by 4, then add the unit digit to this sum and divide this sum by 6 and take remainder.
Ex:- \(\frac{{98037568973}}{6}\)
Sol:- Remainder of \(\frac{{(9 + 8 + 0 + 3 + 7 + 5 + 6 + 8 + 9 + 7) \times 4 + 3}}{6}\)
\(\frac{{ – 1}}{6}\)
= – 1
= 6 – 1
= 5 Answer
Ex:- Find the remainder of \(\frac{{375263908216}}{6}\)
Sol:-
⟶ Division by 4 ⟹ ● Break the number into groups of three from right and calculate remainder of these each term.
● Take the alternating sum of remainders
● Find the remainder of this alternating sum to get the final result.
Ex:- Find remainder when 88643258169436 is divided by 7.
Sol:-
= 4 – (-1) + (-1) – (+1) + 2 = 5 Answer
⟶ Division by 6 ⟹ Take remainder of last three digits that will give the result
Ex:- Find remainder when 8843258169436 is divided by 8
Sol:- Required remainder = remainder of \(\frac{{436}}{8}\) = 4 Answer
⟶ Division by 9 ⟹ Discard all digit which are 9 and also discard all digits of which sum is a multiple of 9 and take sum of remaining digits. Find the remainder of this sum to get the final result.
Ex:- Find remainder when 8843258169442 divided by 9.
Sol:-
⟶ Division by 11 ⟹ Remainder of differece of sum of digits at alternating place
Ex:- Find remainder of 8843258169447 when divided by 11.
Sol:-
Fermat’s Theorem:-
\(\frac{{{a^{p – 1}}}}{p}\) where p is a prime number & (a, p) are co-prime then remainder is 1
Chinese Remainder Theorem:-
\(\frac{N}{{x \times y}}\) where (x,y) are co-prime
then \(\begin{array}{l}
{\mathop{\rm Re}\nolimits} m\left( {\frac{N}{x}} \right) \to a\\
\& {\mathop{\rm Re}\nolimits} m\left( {\frac{N}{y}} \right) \to b
\end{array}\)
The final remainder is smallest number which when divided by x, it leaves remainder a and when divided by y it leaves remainder b.
(Q). Find the last two digit of \({49^{49}}\)
Sol:- Last two digit = Remainder of \(\frac{{{{49}^{49}}}}{{100}}\)
Wilson Theorem:-
\({\mathop{\rm Re}\nolimits} m\frac{{\left( {p – 1} \right)!}}{p} = p – 1\) where p is a prime number
Ex:- Find remainder of \(\frac{{28!}}{{29}}\)
Sol:- Remainder = 28 Answer
Euler Theorem:-
⟶ way to find Euler totient function
● If n is a prime number ⟹ Φ(n) = n – 1
● If n is a power of a prime number ⟹ Φ(pᵏ) = pᵏ – pᵏ⁻¹
● multiplicative property ⟹ If a and b are co-prime, then Φ(a.b) = Φ(a).Φ(b)
● Φ(1) = 1
Ex:-
Φ(1) = 1
Φ(2) = 2 – 1 = 1
Φ(3) = 3 – 1 = 2
Φ(4) = Φ(2²) = 2² – 2 = 2
Φ(4) = Φ(2×2) = Φ(2)×Φ(2) = 1 × 1 = 1 ❌ Wrong
Φ(5) = 5 – 1 = 4
Φ(6) = Φ(2 × 3) = Φ(2) : Φ(3) = (2 – 1)(3 – 1) = 2
Φ(3⁵) = 3⁵ – 3⁴ = 162
Ex:- Find the remainder when 3¹⁰⁰ is divided by 7.
Sol:-
Φ(7) = 7 – 1 = 6
by Euler’s theorem
\(\frac{{{3^{\varphi (7)}}}}{7}\) ⟶ Remainder = 1
\(\frac{{{3^6}}}{7}\) ⟹ Remainder = 1
We need to find remainder of \(\frac{{{3^{100}}}}{7}\)
Ex:- Find the remainder when 7²⁰ is divided by 21.
Sol:-
Φ(21) = Φ(3×7) = (3 – 1)(7 – 1) = 12
∴ \(\frac{{{7^{12}}}}{{21}}\) ⟹ Remainder = 1
To find: Remainder of \(\frac{{{7^{20}}}}{{21}}\)
(Q). Find the remainder when \({32^{{{32}^{32}}}}\) is div
ided by 7.
Sol:-
Φ(7) = 7 – 1 = 6
7 and 32 are co-prime
∴ by Euler’s theorem
\(\frac{{{{32}^{\varphi (n)}}}}{7}\) ⟶ 1 (Remainder)
\(\frac{{{{32}^6}}}{7}\) ⟶ 1
To find: \(\frac{{{{32}^{{{32}^{32}}}}}}{7}\) ⟶ Remainder
So we need to write 32³² as 6Q + R
i.e. we need to find remainder when 32³² is divided by 6.
(Q). Find the remainder when \({27^{{{28}^{29}}}}\) is div
ided by 16.
Sol:-
27 & 16 are co-prime.
Φ(16) = 2⁴ – 2³ = 8
∴ by Euler’s theorem
\(\frac{{{{27}^8}}}{{16}}\) ⟶ 1 (Remainder)
To find: \(\frac{{{{27}^{{{28}^{29}}}}}}{16}\) ⟶ Remainder
So we need to write 28²⁹ in the form 8Q + R & for this find remainder R when 28²⁹ is divided by 8.
● Algebric Remainder Theorem:-
put denominator = 0
ax + b = 0
x = \(\frac{{ – b}}{a}\) ⟶ put this value in numerator which will give remainder as a result.
Ex:- Find remainder when 2x² + 3x + 5 is divided by x – 5.
Sol:-
x – 5 = 0 ⟹ x = 5
put this value in expression:
2×(5)² + 3×5 + 5 = 70 Answer
Ex:- Is (x – 4) a factor of polynomial x² – 3x – 4
Sol:-
put x = 4 in polynomial
4×4 – 3×4 – 4 = 0
Since result is 0 ⟹ (x – 4) is a factor of given polynomial. Answer
● Concept:- 10ⁿ is divided by 6, we always get remainder 4, where n is a natural number.
Ex:- Find the remainder when 10¹ + 10² + 10³ + …………… + 10¹⁰⁰ is divided by 6.
Sol:-
\(\frac{{{{10}^1} + {{10}^2} + {{10}^3} + {{10}^4} + {{……….10}^{100}}}}{6}\)
=\(\frac{{4 \times 100}}{6} = \frac{{4 \times 4}}{6}\) = 4 Answer
● Concept:- If a number is formed by repeating any digit 6 times or a multiple of 6 times then the number is divisible by 3, 7, 11, 13, 37, 39.
Ex:-
111111 ⟶ divisible by 3, 7, 11, 13, 37, 39
999999 ⟶ divisible by 3, 7, 11, 13, 37, 39
(Q). Find the remainder when 5555……..262 times is divided by 37.
Sol:-
\(\frac{{262}}{6} \Rightarrow {\mathop{\rm Re}\nolimits} mainder = 4\)
∴ Remainder of \(\frac{{555…….262times}}{{37}} = {\mathop{\rm Re}\nolimits} m.Of\frac{{5555}}{{37}}\) = 5 Answer
(Q). Find the remainder when 9999…….674 times is divided by 13.
Sol:-