Question

Answer any 3

1. If a and b are two odd positive integers such that a>b, then prove
that one
of the twonumbers +
2
and −
2

Is odd and the other is even.

2. Show that any positive integer is of the form 3q or 3q+1 or 3q+2 for
Some integer q.

3. Find the HCF of the following pairs of numbers using Eucid’sDivisionAlgorithm.

(i) 81 and 237 (ii) 65 and 117 (iii)4052 and 12576

(iv) 56 and 72 (v) 240 and1024

4. 15 pastries and 12 biscuit packets have been donated for a school
fete. These are to be packed in several smaller identical boxes with
the same number of pastries and biscuit packets in each. ow many
biscuit packets and how many pastries will each box contain?

5. 144 cartons of coke cans and 90cartons of Pepsi cans are to be stacked in a
canteen. If each stack is of the same height and is to contain cartons
of the same drink, what would be the greatest number of cartons

each stack would have?

6. Find the HCF and LCM of the following numbers by prime factorization
method and verify that HCF X LCM= product of two given numbers for
(i) and(iii)

(i) 90and144 (Ii)144,180and 192 (iii)96and404

(iv) 391,425and527 (v) 60,84 and108

7. The length,breadth and height of a room are 8m50cm,6m25cm and
4m75cm respectively.Find the length of the longest rod that can measure the dimensions of the room exactly.

8. Can two numbers have15 as their HCF and 175 as theirLCM? Give reasons​​

Answer 1

Answer:

ans 1 We have

a and b are two odd positive integers such that a & b

but we know that odd numbers are in the form of 2n+1 and 2n+3 where n is integer.

so, a=2n+3, b=2n+1, n∈1

Given ⇒ a>b

now, According to given question

Case I:

2

a+b

=

2

2n+3+2n+1

=

2

4n+4

=2n+2=2(n+1)

put let m=2n+1 then,

2

a+b

=2m ⇒ even number.

Case II:

2

a−b

=

2

2n+3−2n−1

2

2

=1 ⇒ odd number.

Hence we can see that, one is odd and other is even.