Question

What is the maximal greatest common divisor of \[6\] different natural numbers written with \[2\] digits?
A. \[13\]
B. \[15\]
C. \[16\]
D. \[18\]

Answer 1

Hint: This question involves the arithmetic operations like addition/ subtraction/ multiplication/ division. We need to know the two-digit prime numbers to make the easy calculation. Also, we need to know how to write the given condition in the question in the form of a mathematical expression to find the perfect answer.

Complete step by step solution:
In this question, we have to find the maximal greatest common factor divisor of \[6\] different natural numbers written with two digits.
The given condition can also be written as, \[{\left( {GCD} \right)_n} \prec 100\] \[ \to \left( 1 \right)\]
Here GCD is nothing but the shortest form of Greatest Common Divisor.
We know that GCD must be a prime number. The prime numbers with two digits are shown below,
 \[2,3,5,7,11,13,17,19\]
In the question, they given that \[n\] must be between \[1\] to \[6\]
If we take \[GCD = 19\] , then the equation \[\left( 1 \right)\] becomes,
 \[\left( 1 \right) \to {\left( {GCD} \right)_n} \prec 100\]
(Here we take \[n\] is \[6\] )
 \[{\left( {GCD} \right)_n} \prec 100 \Rightarrow 19 \times 6 = 114 \succ 100\]
The condition is not satisfied.
If we take \[GCD = 17\] , then we get
 \[\left( 1 \right) \to {\left( {GCD} \right)_n} \prec 100\]
 \[{\left( {GCD} \right)_n} \prec 100 \Rightarrow 17 \times 6 = 102 \succ 100\]
The condition is not satisfied
If we take \[GCD = 13\] , then we get
 \[\left( 1 \right) \to {\left( {GCD} \right)_n} \prec 100\]
 \[{\left( {GCD} \right)_n} \prec 100 \Rightarrow 13 \times 6 = 78 \prec 100\]
The condition is satisfied.
So the correct value of \[GCD = 13\]
So, the final answer is,
The maximal greatest common divisor of \[6\] different natural numbers written with two digits is \[13\] .
So, the option \[A)13\] is the correct answer.
So, the correct answer is “Option A”.

Note: Note that the prime numbers are only divided by \[1\] and itself, we would take only the prime numbers as the greatest common divisor. Also, note that this type of question describes the arithmetic operations like addition/ subtraction/ multiplication/ division. Note that the number \[1\] is an exception, it is not a prime number and it is not a non-prime number.