UFM Additional Further Pure

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isi-entrance 2024 Q16 GCD, LCM, and Coprimality View
Let $n > 1$ be the smallest composite integer that is coprime to $\frac{10000!}{9900!}$. Then
(A) $n \leqslant 100$
(B) $100 < n \leqslant 9900$
(C) $9900 < n \leqslant 10000$
(D) $n > 10000$
isi-entrance 2024 Q24 Quadratic Diophantine Equations and Perfect Squares View
Let $p < q$ be prime numbers such that $p^2 + q^2 + 7pq$ is a perfect square. Then, the largest possible value of $q$ is:
(A) 7
(B) 11
(C) 23
(D) 29
isi-entrance 2026 QB1 Irrationality and Transcendence Proofs View
Let $x$ be an irrational number. If $a , b , c$ and $d$ are rational numbers such that $\frac { a x + b } { c x + d }$ is a rational number, which of the following must be true?
(A) $a d = b c$
(B) $a c = b d$.
(C) $a b = c d$.
(D) $a = d = 0$
isi-entrance 2026 QB4 Quadratic Diophantine Equations and Perfect Squares View
If $n ^ { 2 } + 19 n + 92$ is a perfect square, then the possible values of $n$ may be
(A) - 19
(B) - 8
(C) - 4
(D) - 11
isi-entrance 2026 Q4 10 marks Arithmetic Functions and Multiplicative Number Theory View
Let $S ^ { 1 } = \{ z \in \mathbb { C } | | z \mid = 1 \}$ be the unit circle in the complex plane. Let $f : S ^ { 1 } \rightarrow S ^ { 1 }$ be the map given by $f ( z ) = z ^ { 2 }$. We define $f ^ { ( 1 ) } : = f$ and $f ^ { ( k + 1 ) } : = f \circ f ^ { ( k ) }$ for $k \geq 1$. The smallest positive integer $n$ such that $f ^ { ( n ) } ( z ) = z$ is called the period of $z$. Determine the total number of points in $S ^ { 1 }$ of period 2025. (Hint: $2025 = 3 ^ { 4 } \times 5 ^ { 2 }$)
isi-entrance 2026 Q6 10 marks Quadratic Diophantine Equations and Perfect Squares View
Let $\mathbb { N }$ denote the set of natural numbers, and let $\left( a _ { i } , b _ { i } \right)$, $1 \leq i \leq 9$, be nine distinct tuples in $\mathbb { N } \times \mathbb { N }$. Show that there are three distinct elements in the set $\left\{ 2 ^ { a _ { i } } 3 ^ { b _ { i } } : 1 \leq i \leq 9 \right\}$ whose product is a perfect cube.
isi-entrance 2026 Q18 Modular Arithmetic Computation View
Let $N$ be a 50 digit number. All the digits except the 26th one from the right are 1. If $N$ is divisible by 13, then the unknown digit is
(a) 1 .
(B) 3 .
(C) 7 .
(D) 9 .
jee-main 2017 Q66 Modular Arithmetic Computation View
If $(27)^{999}$ is divided by 7, then the remainder is
(1) 3
(2) 1
(3) 6
(4) 2
jee-main 2019 Q64 GCD, LCM, and Coprimality View
The sum of all natural numbers $n$ such that $100 < n < 200$ and H.C.F.$(91, n) > 1$ is
(1) 3203
(2) 3221
(3) 3121
(4) 3303
jee-main 2020 Q55 Divisibility and Divisor Analysis View
The greatest positive integer $k$, for which $49 ^ { k } + 1$ is a factor of the sum $49 ^ { 125 } + 49 ^ { 124 } + \ldots + 49 ^ { 2 } + 49 + 1$, is
(1) 32
(2) 63
(3) 60
(4) 35
jee-main 2021 Q61 Divisibility and Divisor Analysis View
A natural number has prime factorization given by $n = 2 ^ { x } 3 ^ { y } 5 ^ { z }$, where $y$ and $z$ are such that $y + z = 5$ and $y ^ { - 1 } + z ^ { - 1 } = \frac { 5 } { 6 } , y > z$. Then the number of odd divisors of $n$, including 1 , is:
(1) 12
(2) 6
(3) 11
(4) $6 x$
jee-main 2021 Q67 GCD, LCM, and Coprimality View
Let $A = \{ 2,3,4,5 , \ldots , 30 \}$ and $\sim$ be an equivalence relation on $A \times A$, defined by $( a , b ) \simeq ( c , d )$, if and only if $ad = bc$. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair $( 4,3 )$ is equal to :
(1) 5
(2) 6
(3) 8
(4) 7
jee-main 2021 Q81 Modular Arithmetic Computation View
The total number of two digit numbers $n$, such that $3 ^ { n } + 7 ^ { n }$ is a multiple of 10, is $\underline{\hspace{1cm}}$.
jee-main 2021 Q82 Modular Arithmetic Computation View
If the remainder when $x$ is divided by 4 is 3, then the remainder when $( 2020 + x ) ^ { 2022 }$ is divided by 8 is $\underline{\hspace{1cm}}$.
jee-main 2021 Q86 Modular Arithmetic Computation View
Let $A = \{ n \in N : n$ is a 3-digit number $\}$, $B = \{ 9 k + 2 : k \in N \}$ and $C = \{ 9 k + l : k \in N \}$ for some $l ( 0 < l < 9 )$. If the sum of all the elements of the set $A \cap ( B \cup C )$ is $274 \times 400$, then $l$ is equal to
jee-main 2022 Q63 Modular Arithmetic Computation View
The remainder when $(11)^{1011} + (1011)^{11}$ is divided by 9 is $\_\_\_\_$.
(1) 1
(2) 8
(3) 6
(4) 4
jee-main 2022 Q64 Modular Arithmetic Computation View
The remainder when $3 ^ { 2022 }$ is divided by 5 is
(1) 1
(2) 2
(3) 3
(4) 4
jee-main 2022 Q69 Prime Number Properties and Identification View
Let $R$ be a relation from the set $\{ 1,2,3 \ldots\ldots . , 60 \}$ to itself such that $R = \{ ( a , b ) : b = p q$, where $p , q \geq 3$ are prime numbers\}. Then, the number of elements in $R$ is
(1) 600
(2) 660
(3) 540
(4) 720
jee-main 2023 Q64 Modular Arithmetic Computation View
Let the number $( 22 ) ^ { 2022 } + ( 2022 ) ^ { 22 }$ leave the remainder $\alpha$ when divided by 3 and $\beta$ when divided by 7 . Then $\left( \alpha ^ { 2 } + \beta ^ { 2 } \right)$ is equal to
(1) 20
(2) 13
(3) 5
(4) 10
jee-main 2023 Q65 GCD, LCM, and Coprimality View
If $\operatorname{gcd}(m, n) = 1$ and $1^{2} - 2^{2} + 3^{2} - 4^{2} + \ldots + (2021)^{2} - (2022)^{2} + (2023)^{2} = 1012m^{2}n$ then $m^{2} - n^{2}$ is equal to
(1) 240
(2) 200
(3) 220
(4) 180
jee-main 2023 Q65 Divisibility and Divisor Analysis View
The largest natural number $n$ such that $3 n$ divides 66! is $\_\_\_\_$
jee-main 2023 Q67 Congruence Reasoning and Parity Arguments View
$25 ^ { 190 } - 19 ^ { 190 } - 8 ^ { 190 } + 2 ^ { 190 }$ is divisible by
(1) neither 14 nor 34
(2) 14 but not by 34
(3) 34 but not by 14
(4) both 14 and 34
jee-main 2023 Q85 Modular Arithmetic Computation View
The remainder on dividing $5^{99}$ by 11 is $\underline{\hspace{1cm}}$.
jee-main 2024 Q63 Combinatorial Number Theory and Counting View
Let $A = \{ n \in [ 100,700 ] \cap \mathbb { N } : n$ is neither a multiple of 3 nor a multiple of $4 \}$. Then the number of elements in $A$ is
(1) 290
(2) 280
(3) 300
(4) 310
jee-main 2025 Q12 Modular Arithmetic Computation View
The remainder, when $7 ^ { 103 }$ is divided by 23 , is equal to :
(1) 6
(2) 17
(3) 9
(4) 14

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