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 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 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 Q43 4 marks Combinatorial Number Theory and Counting View
The number of triplets $(a, b, c)$ of integers such that $a < b < c$ and $a, b, c$ are sides of a triangle with perimeter 21 is
(A) 7
(B) 8
(C) 11
(D) 12
isi-entrance 2016 Q43 4 marks Combinatorial Number Theory and Counting View
The number of triplets $(a, b, c)$ of integers such that $a < b < c$ and $a, b, c$ are sides of a triangle with perimeter 21 is
(A) 7
(B) 8
(C) 11
(D) 12
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 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 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 Q1 Congruence Reasoning and Parity Arguments View
Let the sequence $\left\{ a _ { n } \right\} _ { n \geq 1 }$ be defined by
$$a _ { n } = \tan ( n \theta )$$
where $\tan ( \theta ) = 2$. Show that for all $n , a _ { n }$ is a rational number which can be written with an odd denominator.
isi-entrance 2017 Q5 Properties of Integer Sequences and Digit Analysis View
Let $g : \mathbb { N } \rightarrow \mathbb { N }$ with $g ( n )$ being the product of the digits of $n$.
(a) Prove that $g ( n ) \leq n$ for all $n \in \mathbb { N }$.
(b) Find all $n \in \mathbb { N }$, for which $n ^ { 2 } - 12 n + 36 = g ( n )$.
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 Q7 Congruence Reasoning and Parity Arguments View
Let $a , b , c \in \mathbb { N }$ be such that $$a ^ { 2 } + b ^ { 2 } = c ^ { 2 } \text { and } c - b = 1$$ Prove that ( i ) $a$ is odd, ( ii ) $b$ is divisible by 4 ,
(iii) $a ^ { b } + b ^ { a }$ is divisible by $c$.
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 2018 Q25 Quadratic Diophantine Equations and Perfect Squares View
The sum of all natural numbers $a$ such that $a ^ { 2 } - 16 a + 67$ is a perfect square is:
(A) 10
(B) 12
(C) 16
(D) 22.
isi-entrance 2019 Q1 Congruence Reasoning and Parity Arguments View
Prove that the positive integers $n$ that cannot be written as a sum of $r$ consecutive positive integers, with $r > 1$, are of the form $n = 2^{l}$ for some $l \geq 0$.
isi-entrance 2019 Q7 Congruence Reasoning and Parity Arguments View
Let $f$ be a polynomial with integer coefficients. Define $$a_{1} = f(0),\quad a_{2} = f(a_{1}) = f(f(0)),$$ and $$a_{n} = f(a_{n-1}) \quad \text{for } n \geq 3$$ If there exists a natural number $k \geq 3$ such that $a_{k} = 0$, then prove that either $a_{1} = 0$ or $a_{2} = 0$.
isi-entrance 2019 Q12 Congruence Reasoning and Parity Arguments View
A particle is allowed to move in the $XY$-plane by choosing any one of the two jumps:
  1. move two units to right and one unit up, i.e., $( a , b ) \mapsto ( a + 2 , b + 1 )$ or
  2. move two units up and one unit to right, i.e., $( a , b ) \mapsto ( a + 1 , b + 2 )$.
Let $P = ( 30,63 )$ and $Q = ( 100,100 )$. If the particle starts at the origin, then
(A) $P$ is reachable but not $Q$.
(B) $Q$ is reachable but not $P$.
(C) both $P$ and $Q$ are reachable.
(D) neither $P$ nor $Q$ is reachable.

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