Question

What is the largest number that can divide \[410\] , \[751\] , \[1030\] leaving \[7\] as a remainder in each case ?

Answer 1

Hint: We have to find the greatest number by which the three numbers are divided such that we get a remainder of \[7\] for each of the given numbers . We solve this question using the concept of the Greatest Common Factors of the three numbers or simply using the concept of H.C.F. (Highest Common Factor) for the numbers . We will first subtract the remainder from each of the given numbers i.e. we will subtract \[7\] from each term and then we will find the H.C.F. of the numbers . The obtained H.C.F. would be the largest number that can divide all the three numbers leaving \[7\] as a remainder in each case .

Complete step-by-step solution:
Given :
The given numbers are \[410\] , \[751\] , \[1030\] . Also , the remainder left is \[7\] .
Let us consider that
\[A = 410\]
\[B = 751\]
\[C = 103\]
\[r = 7\]
So , as to find the largest number which divides all the three numbers and leave \[7\] remainder . We would subtract 7 from each number , so the numbers becomes as :
\[a = 410 - 7\]
\[a = 403\]
Similarly , we get
\[b = 751 - 7\]
\[b = 744\]
Similarly , we get
\[c = 1030 - 7\]
\[c = 1023\]
Now , we have to find the H.C.F. of the new values \[a\] , \[b\] and \[c\] .
Splitting the numbers into its respective prime factors , we get
\[a = 13 \times 31\]
\[b = 2 \times 2 \times 2 \times 3 \times 31\]
\[c = 3 \times 11 \times 31\]
From the prime factors we can conclude that the highest common factor of the three numbers is \[31\] .
Thus , The largest number that can divide \[410\] , \[751\] , \[1030\] leaving \[7\] as a remainder in each case is \[31\] .

Note: The highest common factor of the numbers is stated as the highest common factors of the given numbers . The factor of the numbers which is common in all the prime factors of the numbers .
Similarly , L.C.M. or the least common multiple is said to be the least common multiple of the given numbers . It is stated as the number which is the multiple of the given numbers and that to the least or we can say the first common multiple .