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NUMBER SYSTEMS

http://nov15.wordpress.com/ Presents QUANT For CAT 2009. NUMBER SYSTEMS. NUMBER SYSTEMS. INTRODUCTION. A number is prime if it is not divisible by any prime number less than it’s square root. Ex: Is 179 a prime number ? Prime Numbers less than 13.3 are 2,3,5,7,11,13

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NUMBER SYSTEMS

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  1. http://nov15.wordpress.com/ Presents QUANT For CAT 2009 NUMBER SYSTEMS

  2. NUMBER SYSTEMS

  3. INTRODUCTION

  4. A number is prime if it is not divisible by any prime number less than it’s square root. Ex: Is 179 a prime number ? Prime Numbers less than 13.3 are 2,3,5,7,11,13 179 is not divisible by any of them, 179 is prime. PRIME NUMBERs

  5. DIVISIBILITY RULES

  6. Test for divisibility by 7: Double the last digit and subtract it from the remaining leading truncated number. If the result is divisible by 7, then so was the original number. Apply this rule over and over again as necessary. Example: 826. Twice 6 is 12. So take 12 from the truncated 82. Now 82-12=70. This is divisible by 7, so 826 is divisible by 7. Test for divisibility by 11: Subtract the last digit from the remaining leading truncated number. If the result is divisible by 11, then so was the first number. Apply this rule over and over again as necessary. Example: 19151 --> 1915-1 =1914 –>191-4=187 –>18-7=11, so yes, 19151 is divisible by 11. Some More Divisibility Rules

  7. Test for divisibility by 13: Add four times the last digit to the remaining leading truncated number. If the result is divisible by 13, then so was the first number. Apply this rule over and over again as necessary. Example: 50661–>5066+4=5070–>507+0=507–>50+28=78 and 78 is 6*13, so 50661 is divisible by 13. Test for divisibility by 17: Subtract five times the last digit from the remaining leading truncated number. If the result is divisible by 17, then so was the first number. Apply this rule over and over again as necessary. Example: 3978–>397-5*8=357–>35-5*7=0. So 3978 is divisible by 17. Test for divisibility by 19: Add two times the last digit to the remaining leading truncated number. If the result is divisible by 19, then so was the first number. Apply this rule over and over again as necessary. Example:101156–>10115+2*6=10127–>1012+2*7=1026–>102+2*6=114 and 114=6*19, so 101156 is divisible by 19. Some more divisibility rules

  8. Remember it! Here is a table using which you can easily remember the previous divisibility rules. Read the table as follows : For divisibility by 7 , subtract 2 times the last digit with the truncated number.

  9. Find the unit’s digit of 71999 (7 to the power 1999) Step 1: Divide the exponent by 4 and note down remainder 1999/4 => Rem = 3 Step 2: Raise the unit’s digit of the  base (7) to the remainder  obtained (3) 73 = 343 Step 3: The unit’s digit of the obtained number is the required answer. 343 => Ans 3 If the remainder is 0, then the unit’s digit of the base is raised to 4 and the unit’s digit of the obtained value is the required answer. If Rem = 0 , Then 74 = XX1 -> Ans 1 Note: For bases with unit’s digits as 1,0,5,6 the unit’s digit for any power will be the 1,0,5,6 itself. Ex: Unit’s digit of 3589494856453 = 6 Unit’s Digit of a Number

  10. We will discuss the last two digits of numbers ending with the following digits in sets : a) 1 b) 3,7 & 9 c) 2,  4, 6 & 8 Last two digits of a number

  11. a) Number ending with 1 : Ex : Find the last 2 digits of 31786 Now, multiply the 10s digit of the number with the last digit of exponent 31786 = 3 * 6 = 18 -> 8 is the 10s digit. Units digit is obviously 1 So, last 2 digits are => 81 Last two digits of a number

  12. b) Number Ending with 3, 7 & 9 Ex: Find last 2 digits of 19266 We need to get this in such as way that the base has last digit as 1 19266 = (192)133 = 361133 Now, follow the previous method => 6 * 3 = 18 So, last two digits are => 81 Last two digits of a number

  13. b) Number Ending with 3, 7 & 9 Remember : 34 = 81 74 = 2401 92 = 81 Last two digits of a number

  14. Ex 2: Find last two digits of 33288 Now, 33288 = (334)72 = (xx21)72 Ten’s digit is -> 2*2 = 04 -> 4 So, last two digits are => 41 Ex 3: find last 2 digits of 87^474 (872)*(874)118 => (xx69) * (xx61)118(6 x 8 = 48) => (xx69)*(81) So, last two digits are 89 Last two digits of a number

  15. c)Ending with 2, 4, 6 or 8 Here, we use the fact that 76 power any number gives 76. We also need to remember that, 242 = xx76 210 = xx24 24even = xx76 24odd = xx24 Last two digits of a number

  16. Ex: Find the last two digits of 2543 2543 = ((210)54) * (23) = ((xx24)54)* 8 = ((xx76)27)*8 76 power any number is 76 Which gives last digits as => 76 * 8 = 608 So last two digits are : 08 Last two digits of a number

  17. Highest power of a number that divides the factorial of another number. What is the highest power of 5 that divides 60!(factorial) Note: N! = N*(N-1)*(N-2)*(N-3)….(2)*(1) Now, Continuously divide 60 with 5 as shown 60/5 = 12, 12/5 = 2 (omit remainders) 2/5 = 0  <- stop at 0 Now add up all the quotients => 12+2+0 = 14 So highest power of 5 that divides 60! is 14. Highest power

  18. Ex: Find Highest power of 15 that divides 100! Here, as 15 is not a prime number  we first split 15 into prime factors. 15 = 5 * 3 Now, find out highest power of 5 that divides 100! and also highest power of 3 that divides 100! . For 5 : 100/5 =20 20/5 = 4 4/5 = 0 So, 20 + 4 + 0 = 24 For 3 : 100/3 = 33 33/3 = 11 11/3 = 3 3/3 = 1 1/3 = 0 So, 33 + 11+ 3 + 1 + 0 = 48 Now, the smallest number of these is taken which will be 24. Highest power

  19. Ex: Find the number of zeroes in 75! This means highest power of 10 which can divide 75! 10 = 5*2 If we consider highest power of 5 which can divide 75! , it’s sufficient. 75/5 =15 15/5 =3 3/5 =0 So, 15+3+0 = 18 So, there are 18 zeroes in 75! Number of zeroes

  20. If the number N can be expressed as a product of prime factors such that N = (pa)*(qb)*(rc) where, p,q,r = prime factors a,b,c = powers to which each is raised Then, No. of factors of N (including 1, N) = (a+1)*(b+1)*(c+1)*…. Number of factors of a number

  21. Even number => Divisible by 2 Odd Number => Not Divisible by 2 Important Results : e x e = e e x o = e o x o = o Even and odd

  22. Download the related exercise here Exercise 1 - Number Systems exercise

  23. If you liked this presentation, do comment on http://nov15.wordpress.com or write to Nicky at nickyswetha20@yahoo.com Let me know!!!

  24. THANK YOU!

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