UFM Additional Further Pure

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jee-advanced 1998 Q14 Divisibility and Divisor Analysis View
14. Number of divisors of the form $4 n + 2 \left( \begin{array} { l l l } n ^ { 3 } & 0 \end{array} \right)$ of the integer 240 is:
(A) 4
(B) 8
(C) 10
(D) 3
jee-advanced 2006 Q8 GCD, LCM, and Coprimality View
8. If $r , s , t$ are prime numbers and $p , q$ are the positive integers such that the LCM of $p , q$ is $r ^ { 2 } t ^ { 4 } s ^ { 2 }$, then the number of ordered pair ( $p , q$ ) is
(A) 252
(B) 254
(C) 225
(D) 224
Sol. (C) Required number of ordered pair (p,q) is $( 2 \times 3 - 1 ) ( 2 \times 5 - 1 ) ( 2 \times 3 - 1 ) = 225$.
jee-main 2013 Q72 Proof of Equivalence or Logical Relationship Between Conditions View
For integers $m$ and $n$, both greater than 1, consider the following three statements : $P : m$ divides $n$, $Q : m$ divides $n ^ { 2 }$, $R : m$ is prime, then
(1) $Q \wedge R \rightarrow P$
(2) $P \wedge Q \rightarrow R$
(3) $Q \rightarrow R$
(4) $Q \rightarrow P$
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 Integer Part or Limit Involving Conjugate Surd Binomial Expansions View
If $\{ \mathrm { p } \}$ denotes the fractional part of the number p , then $\left\{ \frac { 3 ^ { 200 } } { 8 } \right\}$ is equal to
(1) $\frac { 5 } { 8 }$
(2) $\frac { 7 } { 8 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 1 } { 8 }$
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 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 Q83 Modular Arithmetic Computation View
$3 \times 7 ^ { 22 } + 2 \times 10 ^ { 22 } - 44$ when divided by 18 leaves the remainder
jee-main 2022 Q62 Modular Arithmetic Computation View
The remainder when $( 2021 ) ^ { 2023 }$ is divided by 7 is
(1) 2
(2) 3
(3) 4
(4) 5
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 Q63 Modular Arithmetic Computation View
The remainder when $( 2021 ) ^ { 2022 } + ( 2022 ) ^ { 2021 }$ is divided by 7 is
(1) 0
(2) 1
(3) 2
(4) 6
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 Q64 Modular Arithmetic Computation View
The remainder when $7^{2022} + 3^{2022}$ is divided by 5 is
(1) 0
(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 2022 Q72 Arithmetic Functions and Multiplicative Number Theory View
Let $f , g : \mathbb { N } - \{ 1 \} \rightarrow \mathbb { N }$ be functions defined by $f ( \mathrm { a } ) = \alpha$, where $\alpha$ is the maximum of the powers of those primes $p$ such that $p ^ { \alpha }$ divides $a$, and $g ( a ) = a + 1$, for all $a \in \mathbb { N } - \{ 1 \}$. Then, the function $f + g$ is
(1) one-one but not onto
(2) onto but not one-one
(3) both one-one and onto
(4) neither one-one nor onto
jee-main 2022 Q86 GCD, LCM, and Coprimality View
Let $A = \{ n \in N :$ H.C.F.$( n , 45 ) = 1 \}$ and let $B = \{ 2k : k \in \{ 1 , 2 , \ldots , 100 \} \}$. Then the sum of all the elements of $A \cap B$ is $\_\_\_\_$.
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 Q64 Forming Numbers with Digit Constraints View
The total number of 4-digit numbers whose greatest common divisor with 54 is 2 , is $\_\_\_\_$
jee-main 2023 Q65 Find Specific Term from Given Conditions View
Let $f ( x ) = 2 x ^ { n } + \lambda , \lambda \in \mathbb { R } , \mathrm { n } \in \mathbb { N }$, and $f ( 4 ) = 133 , f ( 5 ) = 255$. Then the sum of all the positive integer divisors of $( f ( 3 ) - f ( 2 ) )$ is
(1) 61
(2) 60
(3) 58
(4) 59
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 Q68 Modular Arithmetic Computation View
The remainder when $( 2023 ) ^ { 2023 }$ is divided by 35 is $\_\_\_\_$.
jee-main 2023 Q84 Modular Arithmetic Computation View
The remainder when $19^{200} + 23^{200}$ is divided by 49, is $\_\_\_\_$.

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