Question

Find the greatest numbers that will divide $445,$ $572,$ and $699$ leaving remainder $4,$ $5,$ and $6$ respectively.

Answer 1

Hint: We are given the numbers $445,$ $572,$ and $699$. So now, you have to first subtract the respective remainders from each term, then factorize it and find the HCF of all those new three numbers. You will get the answer.

Complete step-by-step answer:
Have you ever seen the show Fear Factor? It required contestants to face a variety of fear-inducing stunts to win the grand prize of $\mathrm{\$}50000$. At the end of the show, the host would say to the winner, 'Evidently, fear is not a factor for you!' What exactly does that mean? Well, it means that fear doesn't play a part in their actions and decisions. So, then a 'factor' is something that affects an outcome. In mathematics, factors are the numbers that multiply to create another number.
The prime factorization of a number, then, is all of the prime numbers that multiply to create the original number. It would be pretty difficult to perform prime factorization if we didn't first refresh our memory on prime numbers. With that being said, a prime number is a number that can only be divided by one and itself.
The prime factorization of a number is the product of prime factors that make up that number.
So, prime factorization is writing the prime numbers that will multiply together to make a new number as a multiplication problem.
Prime factorization is the product of primes that could be multiplied together to make the original number. Two possible ways of getting the list of the primes include a factor tree and upside down division.
If a prime number occurs more than once in the factorization, it is usually expressed in exponential form to make it look more compact.
So the new numbers we get after subtracting the respective remainders are :
$445-4=441$
$572-5=567$
$699-6=693$
Now let us find the greatest common factor of these three numbers.
So, $441=3\times 3\times 7\times 7$
$567=3\times 3\times 3\times 3\times 7$
$693=3\times 3\times 7\times 11$
The highest common factor (HCF) of all these three numbers is $3\times 3\times 7=63$.
So, the required number is $63$.

Note: Read the question carefully. You should be familiar with the concepts. Also, you must know the HCF. While simplifying, don't miss any term. Also, take care that you simplify it in a simple manner. Many students make mistakes regarding signs so avoid it.