Problems on Numbers

1. 24 is divided into two parts such that 7 times the first part added to 5 times the second part gives 146. The first part is 
a. 11
b. 13
c. 16
d. 17
Correct Option: B
Explanation:
Let the first and second parts be a and 24  a, then
7a+5(24−a)=146
⇒7a+120−5a=146
⇒2a=26 or a = 13

2. The product of two numbers is 120.  The sum of their squares is 289.  The sum of the two numbers is :
a. 20
b. 23
c. 169
d. None
Correct Option: B
Explanation:
Let the number be x and y . Then
(x+y)2=(x2+y2)+2xy=289+2x120
=289+240=529⇒x+y=529−−−√=23

3. The sum of squares of two numbers is 68 and the square of their difference is 36.  The product of the two numbers is 
a. 16
b. 32
c. 58
d. 104
Correct Option:a
Explanation:
Let the numbers be x and y. Then
x2+y2=68&(x−y)2=36
But (x−y)2=36⇒x2+y2−2xy=36
⇒68−2xy=36⇒2xy=32
⇒xy=16

4. The sum of seven numbers is 235.  The average of the first three is 23 and that of the last three is 42.  The fourth number is 
a. 40
b. 126
c. 69
d. 195
Correct Option: A
Explanation:
(23×3+a+42×3)=235⇒a=40

5. Two numbers are such that the ratio between them is 3:5 but if each is increased by 10, the ratio between them becomes 5 : 7, the numbers are
a. 3, 5
b. 7, 9
c. 13, 22
d. 15, 25
Correct Option: D
Explanation:
Let the numbers be 3a and 5a
Then 3a+105a+10=57
⇒7(3a+10)=5(5a+10)⇒a=5
The numbers are 15 & 25

Number System: Factors and Coprimes

A number can be written in its prime factorization format.  For example 100 = 2² x 5² 

Formula 1: The number of factors of a number N = a^p x b^q x c^r ... = (p+1).(q+1).(r+1)...

Example: Find the number of factors of 100.  
Ans: We know that 100 = 2² x 5²
So number of factors of 100 = (2 +1 ).(2 +1) =  9.
Infact the factors are 1, 2, 4, 5, 10, 20, 25, 50, 100

Formula 2: The sum of factors of a number N=ap.bq.cr... can be written as ap+1−1a−1×bq+1−1b−1×cr+1−1c−1...

Example: Find the sum of the factors of 72
Ans: 72 can be written as (23×32). 

Sum of all the factors of 72 = (23+1−12−1×32+1−13−1)= 15 x 13= 195

Formula 3: The number of ways of writing a number as a product of two number =12×[(p+1).(q+1).(r+1)...] (if the number is not a perfect square)

If the number is a perfect square then two conditions arise:

1.  The number of ways of writing a number as a product of two distinct numbers =12×[(p+1).(q+1).(r+1)...−1]

2.  The number of ways of writing a number as a product of two numbers and those numbers need not be distinct= 12×[(p+1).(q+1).(r+1)...+1]