Question

↵For a:
4. Using Euclid's division algorithm, find the HCF of
(i) 405 and 2520 (ii)504 and 1188 (iii) 960 and 1575

5. Show that every positive integer is either even or odd.

6. Show that any positive odd integer is of the form (6m+1) or (6m+3) or (6m+5), where m is some integer.

Answer 1

Dear Student,


Euclid's division lemma:

Let a and b any two positive integers . Then there exist two

unique q and r such that

a = bq + r ,

0 less or equal to ' r ' less than b.


i ) 405 and 2520 , start with the larger integer that is , 2520

Apply the division lemma to 2520 and 405 ,

2520 = 405 × 6 + 90

Since the remainder 90 , we apply the division lemma to

405 and 90

405 = 90 × 4 + 45

90 = 45 × 2 + 0

The remainder has now become zero, so procedure stops.

Therefore ,

HCF( 405 , 2520 ) = 45.

iii) To find HCF of 960 and 1575

1575 = 960 × 1 + 615

960 = 615 × 1 + 345

615 = 345 × 1 + 270

345 = 270 × 1 + 75

270 = 75 × 3 + 45

75 = 45 × 1 + 30

45 = 30 × 1 + 15

30 = 15 × 2 + 0

Now remainder is equal to zero.

Therefore ,

HCF ( 960 , 1575 ) = 15

regards