The remainder on dividing $5^{99}$ by 11 is $\underline{\hspace{1cm}}$.

Let $\alpha = \frac { ( 4 ! ) ! } { ( 4 ! ) ^ { 3 ! } }$ and $\beta = \frac { ( 5 ! ) ! } { ( 5 ! ) ^ { 4 ! } }$. Then :
(1) $\alpha \in \mathrm { N }$ and $\beta \notin \mathrm { N }$
(2) $\alpha \notin \mathrm { N }$ and $\beta \in \mathrm { N }$
(3) $\alpha \in \mathrm { N }$ and $\beta \in \mathrm { N }$
(4) $\alpha \notin \mathrm { N }$ and $\beta \notin \mathrm { N }$

The remainder, when $7 ^ { 103 }$ is divided by 23 , is equal to :
(1) 6
(2) 17
(3) 9
(4) 14

Q82. The remainder when $428 ^ { 2024 }$ is divided by 21 is

The maximum value of $\mathbf { n }$ for which $\mathbf { 4 0 } ^ { \mathbf { n } }$ divides $\mathbf { 6 0 }$ ! Is equal to
(A) 11
(B) 13
(C) 14
(D) 12

(1) Answer the following questions.
(i) Consider an integer $a$. When $a$ is divided by 5, the remainder is 4. So, $a$ can be represented as
$$a = \mathbf { A } \, k + \mathbf { B } \quad ( k \text{ : an integer}).$$
Hence, when $a ^ { 2 }$ is divided by 5, the remainder is $\mathbf { C }$.
(ii) The number written as the three-digit number $120_{(3)}$ in the base-3 system is $\mathbf{DE}$ in the decimal system.
The greatest natural number that can be expressed in three digits using the base-3 system is $\mathbf { F G }$ in the decimal system, and the smallest is $\mathbf { H }$ in the decimal system.
(2) For each of $\mathbf { I }$, $\mathbf { J }$ in the following statements, choose the correct answer from among (0) $\sim$ (3) below.
In the following, let $a$ be an integer and $b$ be a natural number.
(i) "When $a$ is divided by 5, the remainder is 4" is $\mathbf { I }$ for "when $a ^ { 2 }$ is divided by 5, the remainder is $\mathbf { C }$".
(ii) "$b$ satisfies $6 \leqq b \leqq 30$" is $\mathbf { J }$ for "$b$ is a three-digit number in the base-3 system".
(0) a necessary condition but not a sufficient condition
(1) a sufficient condition but not a necessary condition
(2) a necessary and sufficient condition
(3) neither a necessary condition nor a sufficient condition

Let $n$ be a two-digit natural number such that the remainder of $n ^ { 3 }$ divided by 66 is $n$. We are to find the number of such $n$ 's and to find the prime numbers among them.
From the conditions we have
$$n ^ { 3 } = \mathbf { A B } p + n \quad ( 0 < n \leqq \mathbf { C D } ) ,$$
where $p$ is the integer quotient of $n ^ { 3 }$ divided by 66 . This can be transformed into
$$n ( n - 1 ) ( n + 1 ) = \mathrm { AB } p$$
Since either $n - 1$ or $n$ has to be a multiple of $\mathbf { E }$ and either $n - 1 , n$ or $n + 1$ has to be a multiple of $\mathbf { F }$, and furthermore $\mathbf { E }$ and $\mathbf { F }$ are mutually prime, we know that $n ( n - 1 ) ( n + 1 )$ is a multiple of $\mathbf { G }$. (Write the answers in the order $1 < \square < \mathbf { E } < \mathbf { F } < \mathbf { G }$.) Hence one of $n - 1 , n$ and $n + 1$ must be a multiple of $\mathbf{HI}$.
So, since $n \leqq \mathrm { CD }$, the number of $n$ 's where $n - 1$ is a multiple of $\mathbf{HI}$ is $\mathbf{J}$, where $n$ is a multiple of $\mathbf{HI}$ is $\mathbf { K }$, and where $n + 1$ is a multiple of $\mathbf{HI}$ is $\mathbf { L }$.
Thus, the number of $n$ 's is $\mathbf { M N }$ and the prime numbers among them are $\mathbf { O P } , \mathbf { Q R }$, $\mathbf{ST}$, in ascending order.

Answer the following questions.
(1) The prime factorization of 1400 is $\mathbf{A}^{\mathbf{B}} \cdot \mathbf{C}^{\mathbf{D}} \cdot \mathbf{E}$ (give the answers in the order A/C).
(2) The number of the divisors of 1400 is $\mathbf{FG}$.
(3) Let $a$ and $b$ be any two divisors of 1400 satisfying $1 < a < b$. There are $\mathbf { H }$ pairs $( a , b )$ such that $a$ and $b$ are relatively prime and $a b = 1400$. Among them, $a$ and $b$ such that $b - a$ is maximized are
$$a = \mathbf { I } , \quad b = \mathbf { J K L } .$$
(4) For $a = \square$ and $b = \mathbf{JKL}$, consider the equation
We can transform (1) into the following equation:
$$y = \mathbf { M N } x + \frac { \mathbf { O } } { \mathbf { Q } } x - \frac { \mathbf { P } } { \mathbf { Q } } .$$
Therefore, among the pairs of positive integers $x$ and $y$ that satisfy equation (1), the pair such that $x$ is minimized is
$$x = \mathbf { R } , \quad y = \mathbf { S T } .$$

Let $N$ be a positive integer. Both when it is written in base 5 and when it is written in base 9, it is a 3-digit number, but the order of the numerals is reversed. We are to represent $N$ in base 10 (decimal) and in base 4.
Let $N$ be $abc$ in base 5 and $cba$ in base 9. Then we have
$$\mathbf { A } \leq a \leq \mathbf { B } , \quad \mathbf { C } \leq b \leq \mathbf { D } , \quad \mathbf { E } \leqq c \leqq \mathbf { F } \text {. }$$
Since we also have
$$N = \mathbf { G H } a + \mathbf { I } \quad b + c = \mathbf { J K } c + \mathbf { L } \quad b + a ,$$
we obtain
$$b = \mathbf { M } a - \mathbf { N O } c .$$
The $a$, $b$ and $c$ satisfying (1) and (2) are
$$a = \mathbf { P } , \quad b = \mathbf { Q } , \quad c = \mathbf { R } .$$
Thus $N$ expressed in base 10 is $\mathbf { S T U }$, and $N$ expressed in base 4 is $\mathbf { V W X Y }$.

2. For ALL APPLICANTS.

(i) Suppose $x , y$, and $z$ are whole numbers such that $x ^ { 2 } - 19 y ^ { 2 } = z$. Show that for any such $x , y$ and $z$, it is true that
$$\left( x ^ { 2 } + N y ^ { 2 } \right) ^ { 2 } - 19 ( 2 x y ) ^ { 2 } = z ^ { 2 }$$
where $N$ is a particular whole number which you should determine.
(ii) Find $z$ if $x = 13$ and $y = 3$. Hence find a pair of whole numbers ( $x , y$ ) with $x ^ { 2 } - 19 y ^ { 2 } = 4$ and with $x > 2$.
(iii) Hence find a pair of positive whole numbers $( x , y )$ with $x ^ { 2 } - 19 y ^ { 2 } = 1$ and with $x > 1$.
Is your solution the only such pair of positive whole numbers $( x , y )$ ? Justify your answer.
(iv) Prove that there are no whole number solutions $( x , y )$ to $x ^ { 2 } - 25 y ^ { 2 } = 1$ with $x > 1$.
(v) Find a pair of positive whole numbers $( x , y )$ with $x ^ { 2 } - 17 y ^ { 2 } = 1$ and with $x > 1$.
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17. Let $S$ be a set of positive integers, for example $S$ could consist of 3,4 , and 8 .
A positive integer $n$ is called an $S$-number if and only if for every factor $m$ of $n$ with $m > 1$, the number $m$ is a multiple of some number in $S$.
So in the above example, 9 is an $S$-number; this is because the factors of 9 greater than 1 are 3 and 9, and each of these is a multiple of 3 .
Positive integer $n$ is therefore not an $S$-number if and only if
A for every (positive) factor $m$ of $n$ with $m > 1$, there is a number in $S$ which is not a factor of $m$.
B for every (positive) factor $m$ of $n$ with $m > 1$, there is no number in $S$ which is a factor of $m$.
C for every (positive) factor $m$ of $n$ with $m > 1$, every number in $S$ is a factor of $m$.
D for some (positive) factor $m$ of $n$ with $m > 1$, there is a number in $S$ which is not a factor of $m$.
E for some (positive) factor $m$ of $n$ with $m > 1$, there is no number in $S$ which is a factor of $m$.
F for some (positive) factor $m$ of $n$ with $m > 1$, every number in $S$ is a factor of $m$.

A positive integer is called a squaresum if and only if it can be written as the sum of the squares of two integers. For example, 61 and 9 are both squaresums since $61 = 5 ^ { 2 } + 6 ^ { 2 }$ and $9 = 3 ^ { 2 } + 0 ^ { 2 }$.
A prime number is called awkward if and only if it has a remainder of 3 when divided by 4 . For example, 23 is awkward since $23 = 5 \times 4 + 3$.
A (true) theorem due to Fermat states that:
A positive integer is a squaresum if and only if each of its awkward prime factors occurs to an even power in its prime factorisation.
It follows that $5 \times 23 ^ { 2 }$ is a squaresum, since 23 occurs to the power 2 , but $5 \times 23 ^ { 3 }$ is not, since 23 occurs to the power 3 .
Which one of the following statements is not true?

The least common multiple of $b$ and $40$ is $120$.
Accordingly, how many different positive integers $b$ are there?
A) 6
B) 8
C) 10
D) 12
E) 14

If the product $\mathrm{x} \cdot (10!)$ is the square of a positive integer, what is the smallest value that x can take?
A) 21 B) 7 C) 5 D) 10 E) 14

Let p and q be distinct prime numbers such that
$$\begin{aligned} & a = p ^ { 4 } \cdot q ^ { 2 } \\ & b = p ^ { 2 } \cdot q ^ { 3 } \end{aligned}$$
Which of the following is the greatest common divisor of numbers a and b?
A) $p ^ { 5 } \cdot q ^ { 4 }$
B) $p ^ { 4 } \cdot q ^ { 3 }$
C) $p ^ { 3 } \cdot q ^ { 4 }$
D) $p ^ { 2 } \cdot q ^ { 2 }$
E) $p ^ { 2 } \cdot q ^ { 3 }$

$$\begin{aligned} & 2 ^ { x } \equiv 1 ( \bmod 7 ) \\ & 3 ^ { y } \equiv 4 ( \bmod 7 ) \end{aligned}$$
For the smallest positive integers x and y satisfying these congruences, what is the difference $y - x$?
A) 5
B) 4
C) 3
D) 2
E) 1

Let a be a positive integer and $p = a^{2} + 5$. If p is a prime number, which of the following statements are true?
I. a is an even number. II. The remainder when p is divided by 4 is 1. III. $\mathrm{p} - 6$ is prime.
A) I and III B) Only I C) I and II D) Only III E) I, II and III

Let n be a positive integer, and let $S(n)$ denote the set of positive integers that divide n without remainder.
Accordingly, how many elements does the intersection set $S(60) \cap S(72)$ have?
A) 8 B) 9 C) 6 D) 5 E) 4

If a number of the form $7k + 4$ is divisible by 3 without remainder, how many positive integers k less than 21 are there?
A) 8 B) 9 C) 7 D) 6 E) 5

$$\prod _ { n = 1 } ^ { 7 } ( 3 n + 2 )$$
If this number is divisible by $10 ^ { \mathbf { m } }$, what is the maximum integer value that m can take?
A) 2
B) 3
C) 4
D) 5
E) 6

$$\left. \begin{array} { l } 2 ^ { a } \cdot 3 ^ { b } \equiv 0 ( \bmod 12 ) \\ 2 ^ { b } \cdot 3 ^ { a } \equiv 0 ( \bmod 27 ) \end{array} \right\}$$
For positive integers a and b that satisfy both congruences simultaneously, what is the minimum value of the sum $a + b$?
A) 3
B) 4
C) 5
D) 6
E) 7

For $1 < n < 50$, how many integers n are there such that the number of positive divisors is 3?
A) 2
B) 3
C) 4
D) 5
E) 7

Let n be a positive integer. If every prime number p that divides n also divides $p ^ { 2 }$ into n, then n is called a powerful number.
Which of the following is NOT a powerful number?
A) 27
B) 64
C) 72
D) 99
E) 108

n is an integer greater than 1 and
$$\begin{aligned} & 73 \equiv 3 ( \bmod n ) \\ & 107 \equiv 2 ( \bmod n ) \end{aligned}$$
Given this, what is the sum of the possible values of n?
A) 39
B) 41
C) 47
D) 51
E) 54

For a positive integer n, the greatest odd divisor of n is denoted by $\overline{n}$. The terms of the sequence $( a _ { n } )$ are defined for $n = 1,2 , \ldots$ as
$$a _ { n } = \begin{cases} n + 1 , & \text{if } n \equiv 1 ( \bmod 4 ) \\ n - 1 , & \text{if } n \equiv 3 ( \bmod 4 ) \end{cases}$$
Given this, what is the difference $a _ { 18 } - a _ { 12 }$?
A) 2
B) 4
C) 6
D) 8
E) 10