Number System : Co-Primes or Relatively Prime

Co-Prime or Relatively Prime…Hmm
So what do we mean by Co-Primes or Relatively Prime Nos. ?
A set of numbers which do not have any other common factor other than 1 are called co-prime or relatively prime numbers.

This means those numbers whose HCF is 1.

For e.g. : 14 and 15 have no other common factor other than 1 so they are co-prime numbers.
Also called strangers.

Properties of Co-prime Numbers:
~ All prime numbers are co-prime to each other.
~Any 2 consecutive integers are always co-prime.
~ Sum of any two co-prime numbers is always co-prime with their product.
~1 is co-prime with all numbers.
~ a and b (natural numbers) are co-prime only if the numbers 2a-1 and 2b-1 are co-prime.
~ Decimals of Co-Prime Nos. never match

If two numbers a and b are co-prime and the natural number p is divisible by both a and b separately, then the number p is also divisible by a x b.

Decimals of Co-Prime Nos. never match

So when  a number P is divisible by 14 and Q is visible by 15, what are the chances of result being an integer?
Ans. 0 , as 14 and 15 are co-primes.

The addition or difference of two numbers whose denominators are co-prime will never result in an integer
P/14 + Q/15 –> Not an integer
P/14 – Q/15  –> Not an integer

The addition or difference of two numbers whose denominators are not co-prime may result in an integer.
P/3 + Q/9 — > May result in an integer.
For e.g. 10/3 +24/9
Here 10 is not divisible by 3 and 24 is not divisible by 9. But decimal values  are such that on adding it becomes an integer.

Q. Find all the five-digit numbers of the form 46X3Y  that are divisible by 36. ?
This is an interesting question based on the concept of co-prime numbers.  Since 36 is a product of 2 co-prime numbers 4 & 9. So if 46X3Y is divisible by 4 & 9, it will also be divisible by 36.
Lets check the divisibilty of 46X3Y by 4. To be divisible by 4 the last 2 digits of the number must be divisible by 4. Hence Y can be 2 or 6.

If Y is 2, No.  becomes 46X32. Sum of digits = 15+X . For it to be divisible by 9, X= 3
So No is 46332.

If Y is 6, No. Becomes 46X36. Sum of digits = 19+X. For it to be divisible by 9, X = 8.
So No. is 46836.

Solve questions from : Quantitative Aptitude for CAT