# Number System & Theory & Number Properties

This topic is an important and will usually account for about a quarter of the number of questions that typically appear in any B school entrance test - be it XAT or TANCET or CAT or GMAT. Concepts tested include prime numbers, composite numbers, testing whether a given number is prime, co prime or relatively prime numbers, properties of perfect squares, properties of perfect cubes, LCM, HCF or GCD, remainders, test of divisibility, unit digits of numbers, factorials (likely to be tested in XAT than in TANCET), expressing numbers in different bases, number of factors of a positive integer, sum of factors of a positive integer, product of factors of a positive integers, and properties of indices.

If both 11

^{2}and 3^{3}are factors of the number a * 4^{3}* 6^{2}* 13^{11}, what is the smallest possible value of 'a'?- 121
- 3267
- 363
- 33
- 37

**Correct Answer**Choice (3). 363 Correct answer- Explanatory answer
- Divisors; Prime factors
- Hard

Find the largest five digit number that is divisible by 7, 10, 15, 21 and 28.

- 99,840
- 99,900
- 99,960
- 99,990
- 99,970

**Correct Answer**Choice (3). 99,960 Correct answer- Explanatory answer
- Test of divisibility
- Medium

Anita had to multiply two positive integers. Instead of taking 35 as one of the multipliers, she incorrectly took 53. As a result, the product went up by 540. What is the new product?

- 1050
- 540
- 1440
- 1520
- 1590

Let x, y and z be distinct integers. x and y are odd and positive, and z is even and positive. Which one of the following statements cannot be true?

- (x - z)
^{2}y is even - (x - z) y
^{2}is odd - (x - z) y is odd
- (x - y)
^{2}z is even - (x + y)
^{3}z is even

**Correct Answer**Choice (1).

(x - z)^{2}y is even cannot be true Correct answer- Explanatory answer
- Operations on Odd & Even integers
- Medium

- (x - z)
When a number is divided by 36, it leaves a remainder of 19. What will be the remainder when the number is divided by 12?

- 10
- 7
- 19
- 192
- None of these

The sum of the first 100 natural numbers, 1 to 100 is divisible by ____

- 2, 4 and 8
- 2 and 4
- 2
- 100
- None of these

**Correct Answer**Choice (3). The sum is divisible by 2 Correct answer- Explanatory answer
- Sum of AP and divisibility
- Medium

How many different factors does 48 have, excluding 1 and 48?

- 12
- 4
- 8
- 10
- None of these

**Correct Answer**Choice (3).

8 factors excluding 1 and 48. Correct answer- Explanatory answer
- Number of factors
- Medium

10

^{25}- 7 is divisible by _____- 2
- 3
- 9
- Both (2) and (3)

**Correct Answer**Choice (2).

The number is divisible by 3. Correct answer- Explanatory answer
- Test of divisibility
- Easy

Find the G.C.D of 12x

^{2}y^{3}z^{2}, 18x^{3}y^{2}z^{4}, and 24xy^{4}z^{3}- 6xy
^{2}z^{2} - 6x
^{3}y^{4}z^{3} - 24xy
^{2}z^{2} - 24xy
^{3}z^{2} - 18x
^{2}y^{2}z^{3}

- 6xy
What is the value of M and N respectively? If M39048458N is divisible by 8 & 11; where M & N take integer values from 0 to 9, inclusive?

- M = 7; N = 8
- M = 8; N = 6
- M = 6; N = 4
- M = 5; N = 4
- M = 4; N = 6

**Correct Answer**Choice (3).

M = 6; N = 4 Correct answer- Explanatory answer
- Test of divisibility
- Medium

48 students have to be seated such that each row has the same number of students as the others. If at least 3 students are to be seated per row and at least 2 rows have to be there, how many arrangements are possible?

- 4
- 10
- 8
- 7
- 6

**Correct Answer**Choice (4).

7 seating arrangements are possible Correct answer- Explanatory answer
- Counting Methods
- Medium

If two fractions, each of which has a value between 0 and 1, are multiplied together, the product will be ___

- always greater than either of the original fractions
- always less than either of the original fractions
- sometimes greater and sometimes less than either of the original fractions
- remains the same
- never less than either of the original fractions

**Correct Answer**Choice (2).

always less than either of the original fractions Correct answer- Explanatory answer
- Properties of fractions
- Easy
- TANCET 2008

Three Vice Presidents (VP) regularly visit the plant on different days. Due to labour unrest, VP (HR) regularly visits the plant after a gap of 2 days. VP (Operations) regularly visits the plant after a gap of 3 days. VP (sales) regularly visits the plant after a gap of 5 days. The VPs do not deviate from their individual schedules. CEO of the company meets the VPs when all the three VPs come to the plant together. CEO is on leave from January 5

^{th}to January 28^{th}, 2012. Last time the CEO met the VPs on January 3, 2012. When is the next time the CEO will meet all the VPs?- February 6, 2012
- February 7, 2012
- February 8, 2012
- February 9, 2012
- None of the above

**Correct Answer**Choice (3).

February 8, 2012. Correct answer- Explanatory answer
- LCM & Common Multiples
- Moderate
- XAT 2012

How many divisors does 7200 have?

- 20
- 4
- 54
- 32

**Correct Answer**Choice (3).

7200 has 54 factors including 1 and itself. Correct answer- Explanatory answer
- Number of factors
- Moderate

What is the highest power of 7 that will divide 5000! without leaving a remainder? (5000! means factorial 5000)

- 4998
- 714
- 832
- 816

'a' and 'b' are the lengths of the base and height of a right angled triangle whose hypotenuse is 'h'. If the values of 'a' and 'b' are positive integers, which of the following cannot be a value of the square of the hypotenuse?

- 13
- 23
- 37
- 41

**Correct Answer**Choice (2).

The square of the hypotenuse cannot be 23 for integer values of 'a' and 'b'. Correct answer- Explanatory answer
- Properties of perfect squares
- Easy

If x, y, z are chosen from the three numbers, -3, and 2, what is the largest possible value of the expression z

^{2}?- 16
- 24
- 36
- 54

**Correct Answer**Choice (4). 36 Correct answer- Explanatory answer
- Number Properties
- Easy
- TANCET 2008

If ( 0.004 * 10

^{a})( 0.32 * 10^{b}) = 128 * 10^{3}, a + b = ?- 3
- 5
- 6
- 8
- -2

**Correct Answer**Choice (4). a + b = 8 Correct answer- Explanatory answer
- Properties of indices
- Medium
- TANCET 2013

There are 35 steps in a temple. By the time Chithra comes down two steps, Madhu goes up one step. If they start simultaneously and keep their speed uniform, then at which step from the bottom will they meet?

- 9th step
- 12th step
- 13th step
- 8th step
- None of the above

## XAT TANCET Practice Questions - Listed Topic wise

- Number Theory
- Permutation Combination
- Probability
- Inequalities
- Geometry
- Mensuration
- Trigonometry
- Data Sufficiency
- Percentages
- Profit Loss
- Ratio Proportion
- Mixtures & Alligation
- Speed Distance & Time
- Pipes, Cisterns & Work, Time
- Simple & Compound Interest
- Races
- Averages & Statistics
- Progressions : AP, GP & HP
- Set Theory
- Clocks Calendars
- Linear & Quadratic Equations
- Functions
- English Grammar
- General Awareness