Class Discussion - Number Theory

Question 1

A set contains 7 real numbers such that the sum of any three of them is always greater than the sum of the other 4 numbers. Which of the following is definitely true?

		All the numbers can be positive and not necessarily equal
		All the numbers are positive and equal
		All the numbers can be negative and not necessarily equal
		All the numbers are negative and equal
		Such a set does not exist

Question 2

Let $x$, $y$ and $z$ be distinct integers, that are odd and positive. Which one of the following statements cannot be true?
[CAT 2000]

		$xyz^2$ is odd.
		$(x - y)^2 z$ is even.
		$(x + y - z)^2 (x + y)$ is even.
		$(x - y) (y + z) (x + y - z)$ is odd.

Question 3

How many 2 digit numbers exist such that we get a larger number when their digits are reversed?

Answer: 36

Question 4

In a tournament, there are $n$ teams T$_1$ , T$_2$....., T$_n$ with $n > 5$. Each team consists of $k$ players, $k > 3$. The following pairs of teams have one player in common: T$_1$ & T$_2$ , T$_2$ & T$_3$ ,......, T$_{n - 1}$ & T$_n$ , and T$_n$ & T$_1$. No other pair of teams has any player in common. How many players are participating in the tournament, considering all the $n$ teams together?
[CAT 2007]

		$n(k - 1)$
		$k(n - 1)$
		$n(k - 2)$
		$k(k - 2)$
		$(n - 1)(k - 1)$

Question 5

A number is interesting if on adding the sum of the digits of the number and the product of the digits of the number, the result is equal to the number. What fraction of numbers between 10 and 100 (both 10 and 100 included) is interesting?
[XAT 2013]

		0.1
		0.11
		0.16
		0.22
		None of the above

Question 6

Eight variables $-$ $a, b, c, d, e, f, g$ and $h$ $-$ are equal to a different number between 1 and 8 (both inclusive). None of the variables are equal to a value which is the same as their position in the alphabet [e.g. $a \neq 1$, $b \neq 2$, etc.]. The values of any two variables which are adjacent letters in the alphabet, do not have consecutive numbers as values. If $a > f > h > b$ and $d > e > g$, and g is a prime number. If $c + e + g = 7$, what is the value of $h$? (Fill 0 if $h$ cannot be determined)

Answer: 5

Question 7

When $(3 - x)^7$ is expanded, what is the sum of the coefficients of $x^5$, $x^6$ and $x^7$?

		-69
		-139
		169
		None of these

Question 8

What is the remainder when $32^3 + 33^3 + ... + 74^3$ is divided by $105$?

		29
		57
		73
		91