Question

3. An army contingent of 616 members is to march behind an army band of 32 members in
a parade. The two groups are to march in the same number of columns. What is the
maximum number of columns in which they can march?
Use Euclid's division lemma to show that the square of any positive integer is either of
the form 3m or 3m + 1 for some integer m.
[Hint: Letx be any positive integer then it is of the form 37, 3q + 1 or 39+2. Now square
each of these and show that they can be rewritten in the form 3m or 3m + 1.)
Use Euclid's division lemma to show that the cube of any positive integer is of the form
* 9m, 9m + 1 or 9m +8.
The Fundamental Theorem of Arithmetic
your earlier classes, you have seen that any natural number can be written as a
roduct of its prime factors. For instance, 2 = 2,4 = 2 * 2,253 = 11 x 23, and so on.
Jow, let us try and look at natural numbers from the other direction. That is, can any
atural number be obtained by multiplying prime numbers? Let us see.
Take any collection of prime numbers, say 2, 3, 7, 11 and 23. If we multiply
me or all of these numbers, allowing them to repeat as many times as we wish.
e can produce a large collection of positive integers (In fact, infinitely many).
Let us list a few​

Answer 1

Answer:

1.)616 =2³*3*7*11

32. =2power5

HCF=2³

HCF=8

2.)a=bq+r

=3m²

=9m²

=3(3m)²

We have shown that square if any positive integer will be in the form of 3n where m is some integer.

(3m+1)²=3m²+2(3m)(1)+1²

=9m²+6m+1

=3(3m²+2m) +1

We have shown that square of any positive integer will be in the form of

3q+1 where q is some integer