Real numbers

Class 10

The fundamental theorem of arithmetic

• every composite number can be written as the product of primes
• this factorisation is unique apart from the order in which the prime factors occurs.
• for any 2 positive initegers a, b
  • HCF (a,b) * LCM (a,b) = a * b
• factorisation can be done by
  • prime factorisation method
  • factor tree method
• HCF (a,b,c) * LCM (a,b,c) != a * b * c
• LCM (p,q,r) = p* q* r * HCF (p* q * r) / HCF(p* q) * HCF(q* r) * HCF(p* r)
• HCF (p,q,r) = p* q* r* LCM (p* q* r) / LCM(p* q) * LCM(q* r) * LCM(p* r)

Ex.1.2:

1. express as product of primes
2. find lcm and hcf and check the rule
3. check any number can end with a particular digit using the power repitition rule

Revisiting Irrational Numbers

• Theorem 1.3 : If a^2 / p , then a / p, where a is +ve integer
• prove the number is irrational
  1. assume it as rational
  2. write in p / q form where p,q are coprime
  3. sqrt2 = p / q ==> q * sqrt 2 = p
  4. take square on both sides
  5. q * q * 2 = p * p
  6. q * q = p * p / 2
  7. theorem 1.3 ==> 2 divides p, therefore p = 2c, c is an integer
  8. q * q = 4 c * c / 2
  9. q * q = 2 c * c
  10. q * q / 2 = c * c
  11. theorem 1.3 ==> 2 divides q as well
  12. which implies that p and q have 2 as their common factor which contradicts the fact that they are co prime.
  13. this is because of our assumption that sqrt 2 is rational
  14. therefore we conclude that sqrt 2 is irrational.

Ex.1.3

1. prove the given number is irrational

RD Sharma - new

1. Smallest number divisible by 306 and 657 - find LCM - 22338
2. Can 2 numbers have 16 and 380 as HCF and LCM respectively - divide the LCM by HCF, if the remainder is zero, then it is possible - answer "no"
3. 3 bells ring at intervals, when they will ring together - LCM
4. Smallest number increased by 17, exactly divisible by 520 and 468 - find the LCM - then subtract LCM with 17.
5. Find the smallest number when divided by 35,56,91 leaves reminder of 7 - find the LCM , then add 7.
6. Find the largest number which on dividing leaves reminder - find HCF and add the reminder
7. Find the greatest number of 6 digits exactly divisible by 24, 15, 36 - find LCM , then divide the number 999999 by the LCM - subtract the reminder.
8. find the least possible number of tiles - find HCF for the size of the tiles then - area of the courtyard (square) - divide the area of courtyard and area of tile