-  Required number is divisible by 72 as well as by 112, If it is divisible by their L.C.M, which is 1008.
Now, 1008 when divided by 72, Gives quotient = 14.
Required number = 14

To find the least number by which 72 must be multiplied in order to produce a multiple of 112, we need to find the LCM (Least Common Multiple) of 72 and 112, and then divide the LCM by 72 to get the required number.

Step-by-step solution:

• Prime factorization of 72:

72 = 2^3 × 3^2

• Prime factorization of 112:

112 = 2^4 × 7^1

• LCM of 72 and 112:

LCM = 2^4 × 3^2 × 7^1 = 1008

• Divide LCM by 72 to get the required number:

Required number = 1008 ÷ 72 = 14

Therefore, the least number by which 72 must be multiplied in order to produce a multiple of 112 is 14, which is option C.

Some relevant definitions and formulas used in the solution:

• Prime factorization: The process of writing a composite number as a product of its prime factors. For example, the prime factorization of 72 is 2^3 × 3^2.
• LCM (Least Common Multiple): The smallest positive integer that is a multiple of two or more integers. The LCM of a set of integers is the product of the highest powers of all the primes that appear in the factorization of the integers. For example, the LCM of 72 and 112 is 1008.
• Division: The mathematical operation of finding how many times one number is contained within another number. For example, 1008 ÷ 72 = 14.

Note that option E ("None of these") is not the correct answer, as we have shown that the answer is option C.