I hope you understand a terminating decimal.
numbers liable to give a non terminating decimal are 3,6,7,9,11,....
if i have to define it then I would say that a number in denominator would yield a non terminating decimal
if the (number *10)(or factors of 10) is the only way a zero can be in its unit digit than it will Most time result in non terminating decimal. Why i said most I will explain later.
now when we divide a number by another number than for continued division we do the following
suppose we have to evaluate 1/3
the procedure is
multiply and divide by 10
=10/(10*3)
keep aside the 10 in denominator=1/10*(10/3)
=1/10*(3+1/3)----I hope you can understand this.
Now when will this terminate
only when 3*x=y0 where y is 1 or 2(2 remainders we can get by divison by 3 are 1,2)....x is from 1 to 9
thus you can easily see nos like 3,7,9,11,13,17....should all necessarily lead to a non terminating decimal. however nos like 6,12,15 may or may not result (thats why i said MOST)and nos like 2,4,8,10,16,20,25 will always have terminating decimal
why 6,12,15 may or may not terminate depends on if the numerator is divisible by 3 independently or not.
6=3*2
12=3*4
15=3*5
similarly we can see for 14,18 etc
100/12 will not terminate(100 is not divisible by 3
where as 75/12 would certainly terminate=25/4=6.25
If you have understood what i have said above than in order to check if a denominator will yield a terminating/non terminating decimal than just check for all prime factors of the denominator. if numerator does not divide prime factor like 3,7,11 etc it means it will be non terminating decimal.
See if you solve the question asked yourself.
