Let m and n, ($m < n)$be two$2-digit$numbers. Then the total numbers of pairs$(m, n)$, such that gcd$(m, n)$= 6, is ________
Step-by-Step Solution
Key Concept: Write$m=6a$and$n=6b$, then count$two-digit$multiples with$\gcd(a,b)=1$.
Let$m = 6a$,$n = 6$b$(64)$$m < n$$\Rightarrow$$a < b$where a& b are$co-prime$numbers also since m&n are 2 digit nos, so 10$\le m$$\le 99$&10$\le n$$\le 99 i.e. 2$$\le a$$\le 16$&2$\le b$$\le 16 ($∵ a is integer) Now 2$\le a$< b$\le 16$& a& b are$co-prime$∴ if$a = 2$,$b = 3$, 5, 7, 9, 11, 13, 15$a = 3$,$b = 4$, 5, 7, 8, 10, 11, 13, 14, 16$a = 4$,$b = 5$, 7, 9, 11, 13, 15$a = 5$,$b = 6$, 7, 8, 9, 11, 12, 13, 14, 16$a = 6$,$b = 7$, 11, 13$a = 7$,$b = 8$, 9, 10, 11, 12, 13, 15, 16$a = 8$,$b = 9$, 11, 13, 15$a = 9$,$b = 10$, 11, 13, 14, 16$a = 10$,$b = 11$, 13$a = 11$,$b = 12$, 13, 14, 15, 16$a = 12$,$b = 13$$a = 13$,$b = 14$, 15, 16$a = 14$,$b = 15$$a = 15$,$b = 16$64 ordered pairs
Correct Answer: 64