Exercise 1.1

Question.) Use Euclid’s division algorithm to find the HCF of 135 and 225

Answer.) To find the HCF of any two number, we use the Euclid's division lemma which says that:

Given positive integers a and b,

there exist unique integers q and r satisfying a = bq + r,  0 less than and equal to r less than b.

And apply the lemma in Euclid’s division algorithm.

Applying the Euclid’s division algorithm on the numbers 135 and 225, we got the HCF of 135 and 225 as 45

That is 45 is the highest number which can divide 135 and 225 both. 

Question.) Use Euclid’s division algorithm to find the HCF of 196 and 38220

Answer.) To find the HCF of any two number, we use the Euclid's division lemma which says that

Given positive integers a and b,

there exist unique integers q and r satisfying a = bq + r,  0 less than and equal to r less than b.

And apply the lemma in Euclid’s division algorithm.

Applying the Euclid’s division algorithm on the numbers 196 and 38220, we got the HCF of 196 and 38220 as 196

That is 196 is the highest number which can divide 196 and 38220 both. 

Question.) Use Euclid’s division algorithm to find the HCF of 867 and 255

Answer.) To find the HCF of any two number, we use the Euclid's division lemma which says that

Given positive integers a and b,

there exist unique integers q and r satisfying a = bq + r,  0 less than and equal to r less than b.

And apply the lemma in Euclid’s division algorithm.

Applying the Euclid’s division algorithm on the numbers 867 and 255, we got the HCF of 867 and 255 as 3

That is 3 is the highest number which can divide 867 and 255 both.

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

Answer.) We know from below Euclid’s division lemma, that for any positive integers a and b,

There exist unique integers q and r satisfying a = bq + r,  0 less than and equal to r less than b.

Let a be any positive integer.
Dividing a by 6 and using Euclid’s division lemma, we can write a as
a = 6q + 1,
a = 6q + 2,
a = 6q + 3,
a = 6q + 4,
a = 6q + 5,
a = 6q 

But, is should be odd integer according to the question. So, the expression that a can be equal to are
a = 6q + 1,
a = 6q + 3, or
a = 6q + 5, where q is some integer.