The total number of two digit numbers ' n ' , such that 3 n + 7 n is a multiple of 10 , is ___ .

Step-by-step solution

∵ 7 n = ( 10 - 3 ) n = 10 k + ( - 3 ) n 7 n + 3 n = 10 k + ( - 3 ) n + 3 n 10 k + - 3 n + 3 n = 10 k   if n=odd 10 k + 2 . 3 n   if n=even Let n = 2 t ;   t ∈ N ∴ 3 n = 3 2 t = ( 10 - 1 ) t = 10 p + ( - 1 ) t . = 10 p ± 1 ∴ if n = even then 7 n + 3 n will not be multiply of 10 So if n is odd then only 7 n + 3 n will be multiply of 10 ∴ n = 11 , 13 , 15 , … … … . , 99 ∴ Number of odd two digit numbers = 45