This document discusses various methods for determining if a number is divisible by certain integers like 2, 3, 5, 9, and 10. It provides the general forms for 2, 3, and 4 digit numbers. It then explains the divisibility tests for numbers 2 through 10 by examining the ones and tens places and seeing if certain patterns are present. Several examples are worked out applying these divisibility tests. It also explores additional math problems involving sums, products, and relationships between numbers.

1. Javed Alam

2.  The general form of a two digit number is
tens digit x 10 + ones digit
 Example : 43
=4x10 + 3
 59
=5x10 + 9
 63
= 6x10+3

3.  The general form of a 3-digit number is
hundreds digit x 100 + tens digit x 10 +
ones digit.
 Example
 125 = 1x100 + 2 x10 + 5
 234= 2 x 100 + 3 x 10 +4
 554= 5 x 100 + 5 x 10 + 4

4.  The general form of a 4-digit number is
thousands digit x 1000+ hundreds digit x
100 + tens digit x 10 + ones digit.
Example
 4521= 4x1000+5x100+2x10+1
 9845= 9x1000+8x100+4x10+5

5.  For examples
a+3=7 hare a is 4
a4=20 hare a is 5

6. 8 b 2
+ 5 5 a
1 c 7 3
Find the value of
a
b
c

7.  Test of divisibility by 2: a number is divisible
by 2 if the number ends with 0,2,4,6,8.
 General form:
 a two digit number xy can be written as
10x +y in the general form .
 The first term is 10x is divisible by 2 because
10 is divisible by 2. the second term is y is
divisible by 2 if y= 0,2,4,6,8

8.  Test of divisibility by 3: a number is divisible
by 3 if the sum of digits of a number is
divisible by 3.
 General form:
 consider a two digit number xy.
xy=10x + y= (9x+x)+y
The first term is 9x is divisible 3 because 9 is
divisible by 3 and second term x + y isdivisible
by 3 if x+y divisible by 3.
.

9.  Test of divisibility by 5: a number is divisible
by 5 if ones digit is 0 or 5.
 General form:
 consider a two digit number xy.
xy=10x + y
the first term is divisible by 5 because 10 is
divisible by 10 by 5.
The second term is divisible by 5 if y=0 or 5

10.  Test of divisibility by 10: a number is divisible
by 10 if ones digit is 0.
 General form:
 consider a two digit number xy.
xy=10x + y
the first term is divisible by 10 because 10 is
divisible by 10 .
The second term is 0,then the number
becomes 10x and thus divisible by 10.

11.  Test of divisibility by 9: a number is divisible
by 9 if the sum of digits of a number is
divisible by 9.
 General form:
 consider a two digit number xy.
xy=10x + y= (9x+x)+y
The first term is 9x is divisible 9 because 9 is
divisible by 9 and second term x + y is
divisible by 9 if x+y divisible by 9.
.

12.  729 , 3
the sum of the digits of 729 is 7+2+9=18
Since 18 is divisible by 3 , therefore 729 is
divisible by 3.
 709,10
 81954 , 9
 543251 , 3

13.  547 , 10
 625 , 3
 121, 2
 457 ,3

14.  Continue the pattern shown below:
1
1+2+1
1+2+3+2+1
1+2+3+4+3+2+1
1+2+3+4+5+4+3+2+1
a. Calculate the sum of each rows.
Sum of first row= 12
Sum of second row=1+2+1=4=22
Sum of third row= 1+2+3+2+1=9=32
Sum of forth row= 1+2+3+4+3+2+1=16=42

15. b. Calculate the sum of 100th row.

16.  A multiple of eleven I be,
not odd, but even, you see
my digit ( a pair), when multiplied together,
make a cube and a square out of me.

17.  A father gave his son Rs 150. Another father
gave Rs 100 to his son. On adding the
amount that both the sons received, the total
is Rs 150.How is this possible.

18.  Find the smallest integer n such that
5 x 12 x n is the product of three consecutive
integer.

19.  Is the number eleven thousand , eleven
hundred and eleven is divisible by 3

21.  find four numbers whose sum is 45, if the
first number is 8 and first number + 2=
second number -2 = third number x 2 =
fourth number ÷ 2