Question

Find the highest number by which \[101\] and \[137\] can be divided so as to leave a reminder \[5\] in each case.

Answer 1

Hint: To find the highest number by which \[101\] and \[137\] can be divided so as to leave a reminder \[5\] in each case. We will first subtract \[5\] from both the numbers. Then, we will find the H.C.F of the two numbers obtained after subtraction. The number obtained on taking the H.C.F is the answer required.

Complete step by step answer:
As we have to find the greatest number that divides the two, we need to find the highest common factor (H.C.F) of the two numbers obtained after subtraction. The number are given by the difference exactly divisible by the required number are given as:
\[ \Rightarrow 101 - 5 = 96\]
\[ \Rightarrow 137 - 5 = 132\]
To find the greatest number that exactly divides these are given by their H.C.F:
\[ \Rightarrow 96 = 2 \times 2 \times 2 \times 2 \times 2 \times 3\]
\[ \Rightarrow 132 = 2 \times 2 \times 3 \times 11\]
Common numbers \[ = 2 \times 2 \times 3\]
\[\therefore H.C.F = 12\]

Therefore, the highest number by which \[101\] and \[137\] can be divided so as to leave a reminder \[5\] in each case is \[12\].

Additional information: Like H.C.F, least common multiple i.e, LCM is also calculated using the fundamental theorem of arithmetic in which we have to first express the number in terms of multiplication of prime numbers. Then, we find the value of the least common multiple. Then, LCM is determined by taking the highest power of every prime number in the given number.

Note: A factor is a number that divides another number without leaving any remainder. H.C.F of two numbers is the product of common prime factors between the two where a prime number is divisible only by \[1\] and itself. H.C.F of two numbers cannot be greater than either of them. Also, H.C.F can also be called G.C.F which stands for greatest common factor.