# Numbers ( Quantitative Aptitude)

Prime Numbers: A number greater than 1 is called a prime number, if it has exactly two factors, namely 1 and the number itself.
Composite Numbers: Numbers greater than 1 which are not prime are known as composite numbers.
Co-primes: Two numbers a and b are said to be co-primes, if their H.C.F. is 1. e.g., (2, 3), (4, 5), (7, 9), (8, 11), etc. are co-primes,
TESTS OF DIVISIBILITY
Divisibility By 2: A number is divisible by 2, if its unit’s digit is any of 0, 2, 4, 6, 8.
Divisibility By 3: A number is divisible by 3, if the sum of its digits is divisible by 3.
Divisibility By 4: A number is divisible by 4, if the number formed by the last two digits is divisible by 4.
Divisibility By 5: A number is divisible by 5, if its unit’s digit is either 0 or 5.
Divisibility By 6: A number is divisible by 6, if it is divisible by both 2 and 3.
Divisibility By 8: A number is divisible by 8, if the number formed by the last three digits of the given number is divisible by 8.
Divisibility By 9: A number is divisible by 9, if the sum of its digits is divisible by 9.
Divisibility By 10: A number is divisible by 10, if it ends with 0.
Divisibility By 11: A number is divisible by 11, if the difference of the sum of its digits at odd places and the sum of its digits at even places is either 0 or a number divisible by 11.
Divisibility by 12: A number is divisible by 12, if it is divisible by both 4 and 3.
Divisibility By 14: A number is divisible by 14, if it is divisible by 2 as well as 7.
Divisibility By 15: A number is divisible by 15, if it is divisible by both 3 and 5.
Divisibility By 16: A number is divisible by 16, if the number formed by the last 4 digits is divisible by 16.
Divisibility By 24: A given number is divisible by 24, if it is divisible by both 3 and 8.
Divisibility By 40: A given number is divisible by 40, if it is divisible by both 5 and 8.
Divisibility By 80: A given number is divisible by 80, if it is divisible by both 5 and 16.
Note: If a number is divisible by p as well as q, where p and q are co-primes, then the given number is divisible by pq.
If p and q are not co-primes, then the given number need not be divisible by pq, even when it is divisible by both p and q.
(xn – an ) is divisible by (x – a) for all values of n.
(xn – an) is divisible by (x + a) for all even values of n.
(xn + an) is divisible by (x + a) for all odd values of n.
Arithmetic Progression (A.P.) : If each term of a progression differs from its preceding term by a constant, then such a progression is called an arithmetical progression. This constant difference is called the common difference of the A.P.
An A.P. with first term a and common difference d is given by a, (a + d), (a + 2d),(a + 3d),…..
The nth term of this A.P. is given by Tn =a (n – 1) d.
The sum of n terms of this A.P. Sn = n/2 [2a + (n – 1) d] = n/2   (first term + last term).
(1 + 2 + 3 +…. + n) =n(n+1)/2
(12 + 22 + 32 + … + n2) = n (n+1)(2n+1)/6
(13 + 23 + 33 + … + n3) =n2(n+1)2/4
Geometrical Progression (G.P.) : A progression of numbers in which every term bears a constant ratio with its preceding term, is called a geometrical progression.
The constant ratio is called the common ratio of the G.P. A G.P. with first term a and common ratio r is :
a, ar, ar2, ………..
In this G.P. Tn = arn-1
Sum of the n terms, Sn=   a(1-rn)/ (1-r)
Examples
1.     Which of the following are prime numbers?
(i) 241           (ii) 337         (iii) 391           (iv) 571
2.     Find the unit’s digit in the product (2467)163 x (341)72.
3.     Find the unit’s digit in (264)102 + (264)103
4.     What least value must be assigned to * so that the number 197*5462 is divisible by 9 ?
5.     Which digits should come in place of * and \$ if the number 62684*\$ is divisible by both 8 and 5?
6.     Show that 4832718 is divisible by 11.
7.     Is 52563744 divisible by 24?
8.     What least number must be added to 3000 to obtain a number exactly divisible by 19 ?
9.     What least number must be subtracted from 2000 to get a number exactly divisible by 17?
10.                        Find the number which is nearest to 3105 and is exactly divisible by 21.
11.                        Find the smallest number of 6 digits which is exactly divisible by 111.
12.                        On dividing 15968 by a certain number, the quotient is 89 and the remainder is 37. Find the divisor.
13.                        A number when divided by 342 gives a remainder 47. When the same number is divided by 19, what would be the remainder?
14.                        A number being successively divided by 3, 5 and 8 leaves remainders 1, 4 and 7 respectively. Find the respective remainders if the order of divisors be reversed.
15.                        Find the remainder when 231 is divided by 5.
16.                        How many numbers between 11 and 900 are divisible by 7?
17.                        Find the sum of all odd numbers up to 100?
18.                        Find the sum of all 2 digit numbers divisible by 3?
19.                        How many terms are there in 2, 4, 8, 16……1024?
20.                        2 + 22 + 23 + … + 28 =?
21.                        Which of the following are prime numbers?      (i) 241           (ii) 337         (Hi) 391   (iv) 571
Practice questions
1.     The unit digit in the product (784 * 618 * 917 * 463) is:
2.     What is the unit digit in 7105?
3.     What is the unit digit in the product (365 x 659 x 771)?
4.     What is the unit digit in (4137)754?
5.     What is the unit digit in (795 – 358)?
6.     What is the unit digit in {(6374)1793 x(625)317 x(341)491} ?
7.     If the number 481 * 673 is completely divisible by 9, then the smallest whole number in place of * will be:
8.     If the number 517 * 324 is completely divisible by 3, then the smallest whole number in place of * will be:
9.     If the number 97215 * 6 is completely divisible by 11, then the smallest whole number in place or* will be:
10.                        If the number 91876 * 2 is completely divisible by 8, then the smallest whole number in place of * will be:
11.                        Which one of the following numbers is completely divisible by 45?
A.        181560                    B.        381145                     C.        202860         D.        203550
12.                        If the number 42573 * is exactly divisible by 72, then the minimum value of * is:
13.                        Which one of the following numbers is completely divisible by 99 ?
a.     A. 3572404              B.  135792               C.  913464  D.  114345
14.                        If x and y are the two digits of the number 653xy such that this number is divisible by 80, then x+y =?
15.                        If the product 4864 x 9 P 2 is divisible by 12, the value of P is:
16.                        If the number 5 * 2 is divisible by 6, then * =?
17.                        Which of the following numbers is divisible by 24?
A.        35718            B.        63810            C.        537804         D.        3125736
18.                        How many of the following numbers are divisible by 132?
264, 396, 462, 792, 968, 2178, 5184, 6336
19.                        How many 3 digit numbers are divisible by 6 in all?
20.                        If a and b are odd numbers, then which of the following is even?
A.        a + b               B.        a + b + 1        C.        ab       D.        ab + 2
21.                        Which one of the following is a prime number?
A.        119    B.        187    C.        247   D.         None of these
22.                        Which one of the following cannot be the square of a natural number?
A.        30976            B.        81225   C.     42437            D.        20164
23.                        The smallest 3-digit prime number is:
24.                        What smallest number should be added to 4456 so that the sum is completely divisible by 6?
25.                        Which natural number is nearest to 9217, which is completely divisible by 88?
26.                        The largest 4-digit number exactly divisible by 88 is
27.                        The smallest 6-digit number exactly divisible by 111 is :
28.                        On dividing a number by 68, we get 269 as quotient and 0 as remainder. On dividing the same number by 67, what will be the remainder?
29.                        On dividing a number by 56, we get 29 as remainder. On dividing the same number by 8, what will be the remainder?
30.                        In a division sum, the remainder is 0. A student mistook the divisor by 12 instead of 21 and obtained 35 as quotient. What is the correct quotient?
31.                        The sum of the two numbers is 12 and their product is 35. What is the sum of the reciprocals of these numbers?
32.                        On dividing 2272 as well as 875 by 3-digit number N, we get the same remainder. The sum of the digits of N is :
33.                        A boy multiplied 987 by a certain number and obtained 559981 as his answer. If in the answer both 9s are wrong and the other digits are correct, then the correct answer would be:
b.     A.        553681         B.        555181     C.            555681         D.        556581
34.                        Which of the following numbers is divisible by each one of 3, 7, 9 and 11?
A.        639    B.         2079     C.     3791       D.   37911
35.                        A number when divided successively by 4 and 5 leaves remainders 1 and 4 respectively. When it is successively divided by 5 and 4, then the respective remainders will be

36.                        A number was divided successively in order by 4, 5 and 6. The remainders were respectively 2, 3 and 4. The number is