• Natural Numbers: Numbers used for counting (1, 2, 3, ...)
• Even Numbers: Natural numbers divisible by 2
• Odd Numbers: Natural numbers not divisible by 2
• Prime Numbers: Numbers with exactly two factors (1 and the number itself)
• Lowest prime number: 2 (also the only even prime number)
• Whole Numbers: Natural numbers along with zero (0)
• Integers: Natural numbers, their negatives, and zero
• Real Numbers: All numbers that can be represented on the number line

• Rational Numbers: Any number that can be written as p/q, where p and q are integers and q ≠ 0. Denoted by Q.
• Zero (0) is also a rational number.
• Irrational Numbers: Numbers that cannot be expressed as p/q.
• Real Numbers = Rational Numbers + Irrational Numbers

• Terminating (Finite) Decimal Fractions: e.g., 7/8 = 0.875, 21/5 = 4.2
• Non-terminating Decimal Fractions:
  • Recurring (Periodic): e.g., 1/7 = 0.142857142857... = 0.142857
  • Non-recurring (Non-periodic): e.g., 15.2731259629...

• Factors: Numbers multiplied together to get the final number.
• Factors of 12 = 1, 2, 3, 4, 6, and 12.
• All numbers have a factor of one.
• Common Factor: A number that divides each given number completely.
• Common factors of 12 and 18 are 1, 2, 3, 6.
• Co-prime Numbers: Two or more numbers that have no common factor (e.g., 4 and 15).

• The greatest number which divides each of the given numbers exactly.
• Also known as Greatest Common Divisor (G.C.D.).
• Methods to find H.C.F.:
  1. Prime factorisation method
  2. Division method

• Multiples: All numbers that can be divided completely by the given number.
• Multiples of 5 are 5, 10, 15, 20, etc.
• Common Multiples: Numbers that can be exactly divided by each of the given numbers.
• Common multiples of 3 and 4 are 12, 24, etc.
• Least Common Multiple (L.C.M.): The smallest number exactly divisible by each of the given numbers.
• Methods to find L.C.M.:
  1. Prime factorisation method
  2. Division method

• H.C.F. × L.C.M. = Product of the two numbers

• The greatest number that will exactly divide x, y, z = HCF of x, y and z.
• The greatest number that will divide x, y and z leaving remainders a, b and c respectively = HCF of (x−a), (y−b) and (z−c).
• The least number which is exactly divisible by x, y and z = LCM of x, y and z.
• The least number which when divided by x, y and z leaves remainders a, b and c respectively = LCM of (x, y, z) − R, where R = (x−a) = (y−b) = (z−c).
• The least number which when divided by x, y and z leaves the same remainder r in each case = LCM of (x, y, z) + r.
• The greatest number that will divide x, y and z leaving the same remainder in each case = HCF of (x−y), (y−z) and (z−x).

Q1. The four integers next lower than 81, and the four next higher than 81, are written down and added together, this sum is divisible by,

• (a) 7
• (b) 9
• (c) 11
• (d) 13

Q2. How many numbers, between 1 and 300 are divisible by 3 and 5 together?

• (a) 16
• (b) 18
• (c) 20
• (d) 100

Q3. A number when divided by 5 leaves a remainder 3. What is the remainder when the square of the same number is divided by 5?

• (a) 9
• (b) 3
• (c) 0
• (d) 4

Q4. If 5432*7 is divisible by 9, then the digit in place of * is

• (a) 0
• (b) 1
• (c) 6
• (d) 8

Q5. At a college football game, 4/5 of the seats in the lower deck of the stadium were sold. If 1/4 of all the seating in the stadium is located in the lower deck, and if 2/3 of all the seats in the stadium were sold, then what fraction of the unsold seats in the stadium was in the lower deck?

• (a) 3/20
• (b) 1/6
• (c) 1/5
• (d) 1/3

Q6. If a, a + 2 and a + 4 are prime numbers, then the number of possible solutions for a is

• (a) one
• (b) two
• (c) three
• (d) more than three

Q7. The product of two consecutive even numbers is 12768. What is the greater number?

• (a) 110
• (b) 108
• (c) 114
• (d) 112

Q8. What is 131 times 333?

• (a) 46323
• (b) 43623
• (c) 43290
• (d) 44955

Q9. The cost of 3 chairs and 10 tables is 9856. What is the cost of 6 chairs and 20 tables?

• (a) 17227
• (b) 18712
• (c) 19172
• (d) Cannot be determined

Q10. Three friends A, B and C start running around a circular stadium and complete a single round in 24, 36 and 30 seconds respectively. After how many minutes will they meet again at the starting point?

• (a) 12
• (b) 6
• (c) 8
• (d) 15

Q11. If the fractions 2/5, 3/4, 4/5, 5/7 and 6/11 are arranged in ascending order of their values, which one will be the fourth?

• (a) 2/5
• (b) 3/4
• (c) 4/5
• (d) 6/11

Q12. The difference between a two-digit number and the number obtained by interchanging the two digits of the number is 9. The sum of the digits of the number is 15. What is the product of the two digits of the two-digit number?

• (a) 54
• (b) 72
• (c) 56
• (d) Cannot be determined

Q13. If (74)² is subtracted from the square of a number, the answer so obtained is 5340. What is the number?

• (a) 98
• (b) 102
• (c) 104
• (d) 110

Q14. The H.C.F and L.C.M of two numbers are 21 and 4641 respectively. If one of the numbers lies between 200 and 300, then the two numbers are

• (a) 273, 357
• (b) 273, 361
• (c) 273, 359
• (d) 273, 363

Q15. What is the smallest number which when increased by 5 is completely divisible by 8, 11 and 24?

• (a) 264
• (b) 259
• (c) 269
• (d) None of these

Q16. Which is the least number that must be subtracted from 1856, so that the remainder when divided by 7, 12 and 16 will leave the same remainder 4?

• (a) 137
• (b) 1361
• (c) 140
• (d) 172

Q17. Three bells chime at an interval of 18, 24 and 32 minutes respectively. At a certain time they begin to chime together. What length of time will elapse before they chime together again?

• (a) 2 hours 24 minutes
• (b) 4 hours 48 minutes
• (c) 1 hour 36 minutes
• (d) 5 hours

Q18. The L.C.M. of two number is 630 and their H.C.F. is 9. If the sum of numbers is 153, their difference is

• (a) 17
• (b) 23
• (c) 27
• (d) 33

Q19. Suppose you have 108 green marbles and 144 red marbles. You decide to separate them into packages of equal number of marbles. Find the maximum possible number of marbles in each package.

• (a) 4
• (b) 36
• (c) 9
• (d) 12

Q20. I have a certain number of beads which lie between 600 and 900. If 2 beads are taken away the remainder can be equally divided among 3, 4, 5, 6, 7 or 12 boys. The number of beads I have

• (a) 729
• (b) 842
• (c) 576
• (d) 961

Q21. Three men start together to travel the same way around a circular track of 11 kms. Their speeds are 4, 5.5 and 8 kms per hour respectively. When will they meet at the starting point?

• (a) 22 hrs
• (b) 12 hrs
• (c) 11 hrs
• (d) 44 hrs

Q22. A man was engaged for a certain number of days for Rs. 404.30 but being absent for some days he was paid only Rs. 279.90. His daily wages cannot exceed by:

• (a) Rs. 29.10
• (b) Rs. 31.30
• (c) Rs. 31.10
• (d) Rs. 31.41

Q23. The L.C.M. of two numbers is 45 times their H.C.F. If one of the numbers is 125 and the sum of H.C.F. and L.C.M. is 1150, the other number is:

• (a) 215
• (b) 220
• (c) 225
• (d) 235

Q24. There are 4 numbers. The H.C.F. of each pair is 3 and the L.C.M. of all the 4 numbers is 116. What is the product of 4 numbers?

• (a) 9782
• (b) 9396
• (c) 9224
• (d) 9100

Q25. Three wheels can complete respectively 60, 36, 24 revolutions per minute. There is a red spot on each wheel that touches the ground at time zero. After how much time, all these spots will simultaneously touch the ground again?

• (a) 5/2 seconds
• (b) 5/3 seconds
• (c) 5 seconds
• (d) 7.5 seconds

Q26. In a school there are 391 boys and 323 girls. These are to be divided into the largest possible equal classes, so that there are equal number of boys and girls in each class. How many classes are possible?

• (a) 32
• (b) 37
• (c) 42
• (d) 49

Q27. A shopkeeper has three kinds of sugar 184 kg; 230 kg and 276kg. He wants to store it into minimum number of bags to equal size without mixing. Find the size of the bag and the number of bags required.

• (a) 23 kg; 30
• (b) 38 kg; 23
• (c) 46 kg; 15
• (d) 46 kg; 25

Q28. The least number which should be added to 2497 so that the sum is exactly divisible by 5, 6, 4 and 3 is:

• (a) 37
• (b) 13
• (c) 23
• (d) 33

Q29. The H.C.F. of two numbers is 8. Which one of the following can never be their L.C.M.?

• (a) 24
• (b) 48
• (c) 56
• (d) 60

Q30. The sum of two numbers is 462 and their highest common factor is 22. What is the maximum number of pairs that satisfy these conditions?

• (a) 1
• (b) 3
• (c) 5
• (d) 6