Question

Find the smallest number, which when increased by 17, is exactly divisible by both 520 and 468.

Answer 1

Hint: The smallest number which is divisible by a set of given numbers is the least common multiple (LCM) of them.
Find the LCM of the given numbers and subtract 17 from it to get the answer.
The LCM of the given numbers can be found by prime factorization.

Complete step-by-step answer:
If the required number is increased by 17, it becomes divisible by both 520 and 468, which also means that it becomes a multiple of both of them.
Let us find the least common multiple (LCM) of the two numbers by using prime factorization.
  $ 520=52\times 10 $
⇒ $ 520=4\times 13\times 2\times 5 $
⇒ $ 520={{2}^{3}}\times 5\times 13 $
  $ 468=4\times 117 $
⇒ $ 468=4\times 9\times 13 $
⇒ $ 468={{2}^{2}}\times {{3}^{2}}\times 13 $
Now, the LCM is the least number which is a multiple of all of them. Therefore, it must have at least the maximum power of all the prime numbers present in it.
∴ The LCM is $ {{2}^{3}}\times {{3}^{2}}\times 5\times 13=468\times 10=4680 $ .
And the required number is $ 4680-17=4663 $ .
So, the correct answer is “4663”.

Note: There are infinitely many common multiples of a given set of numbers. The LCM is the least of the common multiples and the other common multiples are a multiple of the LCM itself.
The LCM of two numbers $ a $ and $ b $ can also be found out by using the relation: $ LCM(a,b)\times HCF(a,b)=a\times b $ . This relation works for only two numbers.
If LCM of more than two numbers have to be found, then prime factorization is the best approach.
HCF (Highest Common Factor) or GCD (Greatest Common Divisor) of a set of numbers is the greatest number which can divide all the numbers in the set.
There can be more than one common factor of a given set of numbers. The other common factors are the factors of the HCF itself.
1 is a common factor of all the numbers.
Prime Numbers: The numbers which have exactly two factors, i.e. they are not divisible by any other number except for 1 and themselves, are called prime numbers. All other numbers are called composite numbers. 1 is neither prime (it has exactly one factor!), nor composite.