UFM Additional Further Pure

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isi-entrance 2019 Q17 Linear Diophantine Equations View
Let $a _ { 1 } < a _ { 2 } < a _ { 3 } < a _ { 4 }$ be positive integers such that
$$\sum _ { i = 1 } ^ { 4 } \frac { 1 } { a _ { i } } = \frac { 11 } { 6 }$$
Then, $a _ { 4 } - a _ { 2 }$ equals
(A) 11
(B) 10
(C) 9
(D) 8 .
isi-entrance 2019 Q23 Combinatorial Number Theory and Counting View
An examination has 20 questions. For each question the marks that can be obtained are either $-1$ or $0$ or $4$. Let $S$ be the set of possible total marks that a student can score in the examination. Then, the number of elements in $S$ is
(A) 93
(B) 94
(C) 95
(D) 96 .
isi-entrance 2019 Q26 Quadratic Diophantine Equations and Perfect Squares View
The number of integers $n \geq 10$ such that the product $\binom { n } { 10 } \cdot \binom { n + 1 } { 10 }$ is a perfect square is
(A) 0
(B) 1
(C) 2
(D) 3
isi-entrance 2019 Q27 Modular Arithmetic Computation View
Let $a \geq b \geq c \geq 0$ be integers such that $2 ^ { a } + 2 ^ { b } - 2 ^ { c } = 144$. Then, $a + b - c$ equals:
(A) 7
(B) 8
(C) 9
(D) 10 .
isi-entrance 2020 Q7 GCD, LCM, and Coprimality View
Consider a right-angled triangle with integer-valued sides $a < b < c$ where $a, b, c$ are pairwise co-prime. Let $d = c - b$. Suppose $d$ divides $a$. Then
(a) Prove that $d \leq 2$.
(b) Find all such triangles (i.e. all possible triplets $a, b, c$) with perimeter less than 100.
isi-entrance 2020 Q9 Combinatorial Number Theory and Counting View
There are 128 numbers $1,2 , \ldots , 128$ which are arranged in a circular pattern in clockwise order. We start deleting numbers from this set in a clockwise fashion as follows. First delete the number 2, then skip the next available number (which is 3 ) and delete 4 . Continue in this manner, that is, after deleting a number, skip the next available number clockwise and delete the number available after that, till only one number remains. What is the last number left ?
(A) 1
(B) 63
(C) 127
(D) None of the above.
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 Q3 Combinatorial Number Theory and Counting View
Prove that every positive rational number can be expressed uniquely as a finite sum of the form
$$a _ { 1 } + \frac { a _ { 2 } } { 2 ! } + \frac { a _ { 3 } } { 3 ! } + \cdots + \frac { a _ { n } } { n ! } ,$$
where $a _ { n }$ are integers such that $0 \leq a _ { n } \leq n - 1$ for all $n > 1$.
isi-entrance 2021 Q17 GCD, LCM, and Coprimality View
Define $a = p ^ { 3 } + p ^ { 2 } + p + 11$ and $b = p ^ { 2 } + 1$, where $p$ is any prime number. Let $d = \operatorname { gcd } ( a , b )$. Then the set of possible values of $d$ is
(A) $\{ 1,2,5 \}$.
(B) $\{ 2,5,10 \}$.
(C) $\{ 1,5,10 \}$.
(D) $\{ 1,2,10 \}$.
isi-entrance 2021 Q20 Quadratic Diophantine Equations and Perfect Squares View
The number of all integer solutions of the equation $x ^ { 2 } + y ^ { 2 } + x - y = 2021$ is
(A) 5 .
(B) 7 .
(C) 1 .
(D) 0 .
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 Q2 Properties of Integer Sequences and Digit Analysis View
Any positive real number $x$ can be expanded as $x = a _ { n } \cdot 2 ^ { n } + a _ { n - 1 } \cdot 2 ^ { n - 1 } + \cdots + a _ { 1 } \cdot 2 ^ { 1 } + a _ { 0 } \cdot 2 ^ { 0 } + a _ { - 1 } \cdot 2 ^ { - 1 } + a _ { - 2 } \cdot 2 ^ { - 2 } + \cdots$, for some $n \geq 0$, where each $a _ { i } \in \{ 0,1 \}$. In the above-described expansion of 21.1875, the smallest positive integer $k$ such that $a _ { - k } \neq 0$ is:
(A) 3
(B) 2
(C) 1
(D) 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 2022 Q20 Combinatorial Number Theory and Counting View
A $3 \times 3$ magic square is a $3 \times 3$ rectangular array of positive integers such that the sum of the three numbers in any row, any column or any of the two major diagonals, is the same. For the following incomplete magic square
2736
31

the column sum is
(A) 90
(B) 96
(C) 94
(D) 99
isi-entrance 2022 Q23 Linear Diophantine Equations View
The number of triples $( a , b , c )$ of positive integers satisfying the equation $$\frac { 1 } { a } + \frac { 1 } { b } + \frac { 1 } { c } = 1 + \frac { 2 } { a b c }$$ and such that $a < b < c$, equals:
(A) 3
(B) 2
(C) 1
(D) 0
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 Q11 Modular Arithmetic Computation View
Suppose $x$ and $y$ are positive integers. If $4 x + 3 y$ and $2 x + 4 y$ are divided by 7, then the respective remainders are 2 and 5. If $11 x + 5 y$ is divided by 7, then the remainder equals
(A) 0.
(B) 1.
(C) 2.
(D) 3.
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 2023 Q20 Congruence Reasoning and Parity Arguments View
If $[ x ]$ denotes the largest integer less than or equal to $x$, then $$\left[ ( 9 + \sqrt { 80 } ) ^ { 20 } \right]$$ equals
(A) $( 9 + \sqrt { 80 } ) ^ { 20 } - ( 9 - \sqrt { 80 } ) ^ { 20 }$.
(B) $( 9 + \sqrt { 80 } ) ^ { 20 } + ( 9 - \sqrt { 80 } ) ^ { 20 } - 20$.
(C) $( 9 + \sqrt { 80 } ) ^ { 20 } + ( 9 - \sqrt { 80 } ) ^ { 20 } - 1$.
(D) $( 9 - \sqrt { 80 } ) ^ { 20 }$.
isi-entrance 2023 Q29 Combinatorial Number Theory and Counting View
Suppose $f : \mathbb { Z } \rightarrow \mathbb { Z }$ is a non-decreasing function. Consider the following two cases: $$\begin{aligned} & \text { Case 1. } f ( 0 ) = 2 , f ( 10 ) = 8 \\ & \text { Case 2. } f ( 0 ) = - 2 , f ( 10 ) = 12 \end{aligned}$$ In which of the above cases it is necessarily true that there exists an $n$ with $f ( n ) = n$?
(A) In both cases.
(B) In neither case.
(C) In Case 1. but not necessarily in Case 2.
(D) In Case 2. but not necessarily in Case 1.
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

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