The exercise includes the use of Euclids division lemma.
Que1
Use Euclid division algorithm to find HCF of:
1: 135 and 225
2:196 and 38220
3: 867 and 225
Solution:
1: 135 and 225
Here we can see that 225 is greater than 135. Therefore applying Euclid division algorithm we can have,
225 = 135*1 +90
Again remainder 90 is not equal to zero, so we can apply Euclid division lemma for 90 and we get,
135=90*1+45
Again 45 is not equal to zero, repeating the step we have,
90=45*2+0
Here we get remainder zero, and in the last step divisior is 45 therefore we get our hcf.
Hence HCF of 135 and 225 is 45.
2: 196 and 38220
Here 38220 > 196 therefore by applying division lemma and taking 38220 as divisor, we get
38220 = 196*195+0
Remainder is zero.
The HCF is 196.
3: 867 and 225
Here 867 > 225 ,therefore by applying division lemma and taking 867 as divisor we get,
867 =225*3+102
But we don’t get remainder zero therefore by applying Euclid division algorithm and taking 102 as a divisor we get,
225 =102*2+51
Again remainder is not equal to zero so repeating the process we get,
102 = 51*2+0
Now we get the remainder zero.
Hence HCF is 51.
Question 2.
Show that any positive odd integer is of the form 6q+1,or6q+3,or6q+5, where q is some integer.
Solutions: let us assume that a be any positive integer and b = 6 , now applying Euclid division algorithm we have a = 6q+r, where q>=0 and 0<=r<6
Now we will take the values of r one by one.
If r=0 then a=6q,
Similarly if we take values of r =1,2 3,4,5 we will have values of a as
6q+1,6q+2,6q+3,6q+4 and 6q+5 respectively.
Now a positive integer and either be even or odd. If a =6q, 6q+2,6q+4 ,then a is an even number .
Therefore any positive odd integer is of the form 6q+1,6q+3,6q+5.
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 maxim number of columns in which they can march.
Solutions :
We have,
Number of army contingent = 616
Number of army band members = 32
Maximum number of columns in which they can march if the two groups have to march in a same column can be find out as the HCF of (616,32).
Now by applying Euclid division algorithm to find out their hcf we have
616 > 32, so taking 616 as a divisor we will get,
616= 32*19+8
Here remainder is not equal to zero. So taking 32 as divisor and applying division lemma we get,
32 =8*4+0
Here we have remainder zero therefore HCF is 8.
Hence the maximum number of columns in which they can march is 8.
