UFM Additional Further Pure

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isi-entrance 2016 Q3 4 marks Properties of Integer Sequences and Digit Analysis View
The last digit of $( 2004 ) ^ { 5 }$ is
(A) 4
(B) 8
(C) 6
(D) 2
isi-entrance 2016 Q30 4 marks Prime Number Properties and Identification View
Suppose $a, b$ and $n$ are positive integers, all greater than one. If $a^n + b^n$ is prime, what can you say about $n$?
(A) The integer $n$ must be 2
(B) The integer $n$ need not be 2, but must be a power of 2
(C) The integer $n$ need not be a power of 2, but must be even
(D) None of the above is necessarily true
isi-entrance 2016 Q52 4 marks Modular Arithmetic Computation View
$N$ is a 50 digit number. All the digits except the 26th from the right are 1. If $N$ is divisible by 13, then the unknown digit is
(A) 1
(B) 3
(C) 7
(D) 9
isi-entrance 2016 Q52 4 marks Modular Arithmetic Computation View
$N$ is a 50 digit number. All the digits except the 26th from the right are 1. If $N$ is divisible by 13, then the unknown digit is
(A) 1
(B) 3
(C) 7
(D) 9
isi-entrance 2016 Q61 4 marks Quadratic Diophantine Equations and Perfect Squares View
If $n$ is a positive integer such that $8n + 1$ is a perfect square, then
(A) $n$ must be odd
(B) $n$ cannot be a perfect square
(C) $2n$ cannot be a perfect square
(D) none of the above
isi-entrance 2016 Q61 4 marks Quadratic Diophantine Equations and Perfect Squares View
If $n$ is a positive integer such that $8 n + 1$ is a perfect square, then
(A) $n$ must be odd
(B) $n$ cannot be a perfect square
(C) $2 n$ cannot be a perfect square
(D) none of the above
isi-entrance 2016 Q67 4 marks Properties of Integer Sequences and Digit Analysis View
The digit in the unit's place of the number $1! + 2! + 3! + \ldots + 99!$ is
(A) 3
(B) 0
(C) 1
(D) 7
isi-entrance 2016 Q72 4 marks Arithmetic Functions and Multiplicative Number Theory View
Let $d_1, d_2, \ldots, d_k$ be all the factors of a positive integer $n$ including 1 and $n$. If $d_1 + d_2 + \ldots + d_k = 72$, then $\frac{1}{d_1} + \frac{1}{d_2} + \cdots + \frac{1}{d_k}$ is:
(A) $\frac{k^2}{72}$
(B) $\frac{72}{k}$
(C) $\frac{72}{n}$
(D) none of the above
isi-entrance 2017 Q6 GCD, LCM, and Coprimality View
In the Mathematics department of a college, there are 60 first year students, 84 second year students, and 108 third year students. All of these students are to be divided into project groups such that each group has the same number of first year students, the same number of second year students, and the same number of third year students. What is the smallest possible size of each group?
(A) 9
(B) 12
(C) 19
(D) 21.
isi-entrance 2017 Q6 Quadratic Diophantine Equations and Perfect Squares View
Let $p _ { 1 } , p _ { 2 } , p _ { 3 }$ be primes with $p _ { 2 } \neq p _ { 3 }$, such that $4 + p _ { 1 } p _ { 2 }$ and $4 + p _ { 1 } p _ { 3 }$ are perfect squares. Find all possible values of $p _ { 1 } , p _ { 2 } , p _ { 3 }$.
isi-entrance 2018 Q6 Combinatorial Number Theory and Counting View
A number is called a palindrome if it reads the same backward or forward. For example, 112211 is a palindrome. How many 6-digit palindromes are divisible by 495?
(A) 10
(B) 11
(C) 30
(D) 45
isi-entrance 2018 Q17 Quadratic Diophantine Equations and Perfect Squares View
The number of pairs of integers $( x , y )$ satisfying the equation $x y ( x + y + 1 ) = 5 ^ { 2018 } + 1$ is:
(A) 0
(B) 2
(C) 1009
(D) 2018.
isi-entrance 2021 Q3 GCD, LCM, and Coprimality View
The number of ways one can express $2 ^ { 2 } 3 ^ { 3 } 5 ^ { 5 } 7 ^ { 7 }$ as a product of two numbers $a$ and $b$, where $\operatorname { gcd } ( a , b ) = 1$, and $1 < a < b$, is
(A) 5 .
(B) 6 .
(C) 7 .
(D) 8 .
isi-entrance 2021 Q22 Quadratic Diophantine Equations and Perfect Squares View
For a positive integer $n$, the equation $$x ^ { 2 } = n + y ^ { 2 } , \quad x , y \text { integers} ,$$ does not have a solution if and only if
(A) $n = 2$.
(B) $n$ is a prime number.
(C) $n$ is an odd number.
(D) $n$ is an even number not divisible by 4 .
isi-entrance 2022 Q5 Arithmetic Functions and Multiplicative Number Theory View
For any positive integer $n$, and $i = 1, 2$, let $f_i(n)$ denote the number of divisors of $n$ of the form $3k + i$ (including 1 and $n$). Define, for any positive integer $n$, $$f(n) = f_1(n) - f_2(n)$$ Find the values of $f\left(5^{2022}\right)$ and $f\left(21^{2022}\right)$.
isi-entrance 2022 Q9 Modular Arithmetic Computation View
Suppose the numbers 71, 104 and 159 leave the same remainder $r$ when divided by a certain number $N > 1$. Then, the value of $3 N + 4 r$ must equal:
(A) 53
(B) 48
(C) 37
(D) 23
isi-entrance 2022 Q19 Modular Arithmetic Computation View
The number of positive integers $n$ less than or equal to 22 such that 7 divides $n ^ { 5 } + 4 n ^ { 4 } + 3 n ^ { 3 } + 2022$ is
(A) 7
(B) 8
(C) 9
(D) 10
isi-entrance 2023 Q1 Properties of Integer Sequences and Digit Analysis View
Determine all integers $n > 1$ such that every power of $n$ has an odd number of digits.
isi-entrance 2023 Q4 Divisibility and Divisor Analysis View
Let $n _ { 1 } , n _ { 2 } , \cdots , n _ { 51 }$ be distinct natural numbers each of which has exactly 2023 positive integer factors. For instance, $2 ^ { 2022 }$ has exactly 2023 positive integer factors $1,2,2 ^ { 2 } , \cdots , 2 ^ { 2021 } , 2 ^ { 2022 }$. Assume that no prime larger than 11 divides any of the $n _ { i }$'s. Show that there must be some perfect cube among the $n _ { i }$'s. You may use the fact that $2023 = 7 \times 17 \times 17$.
isi-entrance 2023 Q4 Divisibility and Divisor Analysis View
The number of consecutive zeroes adjacent to the digit in the unit's place of $401 ^ { 50 }$ is
(A) 3.
(B) 4.
(C) 49.
(D) 50.
isi-entrance 2023 Q15 Divisibility and Divisor Analysis View
Let $n$ be a positive integer having 27 divisors including 1 and $n$, which are denoted by $d _ { 1 } , \ldots , d _ { 27 }$. Then the product of $d _ { 1 } , d _ { 2 } , \ldots , d _ { 27 }$ equals
(A) $n ^ { 13 }$.
(B) $n ^ { 14 }$.
(C) $n ^ { \frac { 27 } { 2 } }$.
(D) $27 n$.
isi-entrance 2024 Q11 Prime Number Properties and Identification View
Let $n \geqslant 1$. The maximum possible number of primes in the set $\{n+6, n+7, \ldots, n+34, n+35\}$ is
(A) 7
(B) 8
(C) 12
(D) 13
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 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 .

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