NUMBER SYSTEM

 NUMBER SYSTEM 

Face Value:- its own value in a numeral, at whatever place it may be.

Place value / local value: place of digits in numerals.

Example :- 

• What is the difference in the place value of 5 in the numeral 754853?

     Ans:- 50000 - 50 = 49950

Various Types of Numbers:-

1. Natural Numbers:- all counting numbers 

2. Whole Numbers:- all counting numbers and 0

3. Integers:- all counting numbers,  0 and negative counting numbers. 

4. Co-prime Numbers:- two natural numbers are said to be co-prime,  if their HCF is 1.

[(Divisor) × (Quotient)] + Reminder = Dividend 

Examples:-

1. A number is divided by 296 leaves 75 as reminder.  When the same number is divided by 37, the reminder will be :

Solution:

Let x = 296q + 75 = 37 × 8q + 37 × 2 + 1

  = 37(8q + 2) + 1

So, the reminder is 1.

2. What least number must be subtracted from 13601, so that reminder is divisible by 87? (Ans.29)

3. What smallest should be added to 4456, so that sum is completely divisible by 6? (Ans.2)

4. The smallest 6- digit number exactly divisible by 111 is :-  (Ans.100011)

5. The largest 5-digit number exactly divisible by 91 is :-  (Ans.99918)

6. Find the total number of prime factors in the product (4¹¹ × 7⁵ × 11²)

Solution:

  4¹¹ × 7⁵ × 11² = (2 × 2)¹¹ × 7⁵ × 11² = 2²² × 7⁵ × 11² So, required no. of prime factors = 22 + 5 + 2 = 29

7. What is the unit digit in the product (684 × 759 × 413 × 676) ? (Ans. 8)

8. How many two digit numbers are divisible by 3 or 5?

Solution: 

   No. of two digits are 99 - 9 = 90
No. of. two digit numbers divisible by 3 or 5 = 90/3 + 90/5 - 90/15 = 30 + 18 - 6 = 42 (Ans)

9. The largest three digit number which gives the same remainder 2 when divided by 3, 5 and 9 is ----?

Solution:

LCM of 3, 5, 9 = 45

Largest three digit number = 999

On dividing 999 by 45 , reminder will be 9.

Then divisible number = 999 - 9 = 990

Given, in each case reminder = 2

So, required number = 990 + 2 = 992.

10. What is the remainder when the product of 335, 608 and 853 is divided by 13?

Solution:

     Remainder of (335 ÷ 13) = 10

     Remainder of (608 ÷ 13) = 10

     Remainder of (853 ÷ 13) = 8

According to question, remainder of (product of 335, 608 and 853) ÷ 13 = remainder of (10 × 10 × 8) ÷ 13 

So answer is remainder of (800 ÷ 13) = 7

• Sum of 'n' no. of even numbers = n(n + 1)
• Sum of 'n' no. of odd numbers = n²

Trick :-  x × y/z = (x + 1) - (z - y) / z

Example :- 5 × 6 / 7 = (5 + 1) - (7 - 6) / 7 = 6 - 1/7.

• Rule for divisibility by 7 : - Double the last digit & subtract it from the rest of the number and the answer is '0' or divisible by 7.
• Perfect number :- If the sum of divisors of a no. N excluding itself is equal to N, then N is called a perfect number.

     Ex:- 6, 28, 496, 8128, etc.

NOTE :- The sum of the reciprocals of the divisors of a perfect number including that of its own is always equal to '2'.

Important Facts

1. (xⁿ - aⁿ) is divisible by (x - a) for all values of n.
2. (xⁿ - aⁿ) is divisible by (x + a) for all even values of n.
3. (xⁿ + aⁿ) is divisible by (x + a) for all odd values of n.

Example:-

1. What will be the reminder when 17²⁰⁰ is divided by 18?
Solution:
 Here n = 100; an even
 (17²⁰⁰ - 1²⁰⁰) is divisible by 17 + 1 = 18
On dividing 17²⁰⁰ by 18, we get 1 as reminder. 
2. What will be the reminder when 25²⁵ is divided by 26?
Solution:
Here n = 25 an odd
(25²⁵ + 1²⁵) is divided by 25 + 1 = 26
On dividing 25²⁵ by 26, we get 25 (26 - 1) as reminder. 

3. Find the remainder when 2³¹ is divided by 5? (Ans. 3)

Series Problems 

1. 1 + 2 + 3 +....... + n = n(n + 1) / 2 (sum of natural numbers)
2. 1² + 2² + 3² +....... + n² = n(n + 1)(2n + 1) / 6 (sum of squares of natural numbers)
3. 1³ + 2³ + 3³+....... + n³ = [n(n + 1) / 2]² =             n²(n + 1)² / 4 (sum of cubes of natural numbers)

Example 

1. 2² + 4² + 6² + ............+ 20² = ? (Ans.1540)

2. If 1³ + 2³ + 3³ +.......+9³ = 2025, then value of (0.11)³ + (0.22)³ + (0.33)³+......+ (0.99)³ is close to ‐‐‐? (Ans. 2.695)

Do it yourself / Previous years Questions 

1. What is the number when multiplied by 16 increases by 540?

2. How many numbers from 1 to 20 are either a multiple of 3 or 5 ?

3. If the unit digit of (433 × 456 × 43N) is (N + 2), then what is the value of N? (Ans. 6)

4. When a two-digit number is multiplied by the sum of its digits,  the product is 424. When the number obtained by interchanging its digits is multiplied by sum of the digits,  the result is 280. The sum of the digits of the given number is :- ? (Ans. 8)

5. The difference between a number and the square root of the number is 2. The number is-----? (Ans.4)

6. If a number is as much greater than 31 as it is less than 75, then the number is ------. (Ans. 53)

7. If the number 647592 is divisible by 88 and if the digits are rearranged in increasing order,  then the new number thus formed will be divisible by: ? (Ans.3)

8. In an office,  there are 108 tables and 132 chairs.  If 1/6 of the tables and 1/4 of the chairs are broken.  How many people can work in the office,  if each person requires one table and one chair? (Ans.90)

9. 57 sweets are distributed among 10 children such that each girl gets 6 sweets and each boy gets 5 sweets. The number of boys is ----? (Ans.3)

10. The students of a class donated a sum of Rs.2209. If each student donated as many rupees as the number of students in the class, then the number of students in the class is : ? (Ans.47)

11. What is the sum of first 20 terms of the following series?

         1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 + ...... (Ans. 3080)

12. What is the value of 14³ + 16³ + 18³ + ......+ 30³ ?

13. The largest number of four digit that is exactly divisible by 17 and 36 is ----? (Ans. 9792)