Numbers Formulae Some important formulas

Numbers Formulas

Some important formulas:

Mr.Maths,

(a + b)(a - b) = a 2-b 2.

(a + b)2 = (a 2 + b 2 + 2ab).

(a - b)2 = (a 2 + b 2 - 2ab).

(a + b + c)2 = a 2 + b 2 + c 2 + 2 (ab+bc+ca).

(a 3 + b 3) = (a+b)(a 2 -ab+ b 2).

(a 3- b 3) = (a-b)(a 2 +ab+ b 2).

(a 3 + b 3+ c 3- 3abc) 

= (a+b+c)(a 2+b 2+ c 2-ab-bc-ca).

When a + b + c = 0; then a 3 + b 3 + c 3 = 3abc.

Properties of Numbers

A three digit number formed by repeating a one digit number thrice, is always divisible by 111.

Since 111=3 × 37, such numbers are divisible by 3, 37, 111.

Ex:

222, 777, 888 etc, are divisible by 3, 37, 111.

A four digit number formed by repeating a two digit number, is always divisible by 101.

Ex: 

2525, 3636, 8383 etc, are divisible by 101.

The number zero surrounded by the same two digit number on both sides. Any such five digit number is always divisible by 1001.

Since 1001 = 7 × 11 × 13. Such numbers are divisible by 7, 11, 13 and 1001.

Ex: 52052, 68068, 93093, 85085.

A six digit number formed by repeating a three digit number is always divisible by 1001.

Since 1001 = 7 × 11 × 13. Such numbers are divisible by 7,11,13 also. `

Ex:

 436436, 850850, 789789, 123123.

If sum of digits of a natural number is subtracted from the number itself, the resulting number is always divisible by 9.

Ex: 143: Sum of digits = 1 + 4 + 3 = 8

Now 143 - 8 = 135, which is divisible by 9.

Sum of two digit number and number formed by interchanging its digits, is always divisible by 11.

Ex1:

 38 + 83 = 121 which is divisible by 11.

Ex2:

 98 + 89 = 187 which is divisible by 11.

Digits of a two digit number and reversed. Difference between the original number and new number is always divisible by 9.

Ex: 

71 - 17 = 54, which is divisible by 9.

Digits of three digits number are reversed. Difference between the original number and new number is always divisible by 99.

Ex: 

731 - 137 = 594, which is divisible by 99.

Difference between square of a two digit number (xy) and square of a number formed by interchanging the digits (yx), is always divisible by 99.

Ex:

 21 2 - 12 2 = 441 - 144 = 297 = 99 × 3.

Test of Divisibility:

Divisibility By 2:

A number is divisible by 2 if its unit digit is any of of ,2,4,6,8.

Ex: 58694 is divisible by 2, while 86945 is not divisibleby 2.

Divisibility By 3:

A number is divisible by 3 only when the sum of its digits is divisible by 3.

Ex: In the number 695421, the sum of digits = 27, which is divisible by 3, then 695421 is divisible by 3.

Divisibility By 4:

A number is divisible by 4 is the last two digits is divisible by 4.

Ex1:

 6879376 is divisible by 4, since 76 is divisible by 4.

Ex:2 496138 is not divisible by 4, since 38 is not divisible by 4.

Divisibility By 5:

A number is divisible by 5 when its unit digit is 0 or 5.

Ex:

 76895 and 68790 are divisible with 5.

Divisibility By 8:

A number is divisible by 8 if the number formed by hundreds tens and units digit of the given number is divisible by 8.

Ex: In the number 16789352 the number formed by last three digits 352 is divisible by 8 then whole number is divisible by 8.

Divisibility By 9:

A number is divisible by 9 only when the sum of its digits is divisible by 9.

Ex: 

In number 246591, the sum of digits = 27, which is divisible by 9, then whole number is divisible by 9.

Divisibility By 10:

A number is divisible by 10 only when its unit digit is 0.

Ex: Each of the numbers 76890 and 68790 is divisible by 10.

Divisibility By 11:

A number is divisible by 11 if the difference between 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.

Ex1:

Take the number 29435417.

= (Sum of its digits at odd places) - (Sum of its digits at even places).

= ( 7 + 4 + 3 + 9 ) - ( 1 + 5 + 4 + 2 )

= ( 23 - 12)

= 11.

Ex2:

Consider the number 57463822.

= (Sum of its digits at odd places) - (Sum of its digits at even places).

= ( 2 + 8 + 6 + 7 ) - ( 2 + 3 + 4 + 5 )

= ( 23 - 14 )

= 9

which is not divisible by 11. Then 57463822 is not divisible by 11.