Maths help 10 points best answer

please can you help me with this question
the smallest positive integer value for n for which 168n is a multiple of 324 WITHOUT using a calculator
also tell me how you obtained ur answer thanks this is a serious question as i have my exam tommorw please help me out

prime factorisation of 168 = 2 x 2 x 2 x 3 x 7
= (2 x 2 x 3) x 2 x 7
prime factorisation of 324 = 2 x 2 x 3 x 3 x 3 x 3
= (2 x 2 x 3) x 3 x 3 x 3
so HCF = (2 x 2 x 3) = 12
so LCM = 12 x 2 x 7 x 3 x 3 x 3 = 4536

so n = 4536 divided by 168 = 27

and 168n then equals 4536 which is 14 x 324

Tricky without a calculator but not impossible.

we need a number such that
168*n
-------- is an integer and this integer should be minimum
324
This is equal to
14*n
-------
27
we need to select such that n is minimum and divisible by 27
The minimum number value of n=27
The number is 14*27=378

168n = 324m
84 n = 162m
42n = 81m
14n = 27m
n = 27/14 *m
m = multiple of 14 for n to be a positive integer
when m = 14 , n =27 is the smallest positive integer value for n for which 168n is a multiple of 324

prime factorisation of 168 = 2 x 2 x 2 x 3 x 7
= (2 x 2 x 3) x 2 x 7
prime factorisation of 324 = 2 x 2 x 3 x 3 x 3 x 3
= (2 x 2 x 3) x 3 x 3 x 3
so HCF = (2 x 2 x 3) = 12
so LCM = 12 x 2 x 7 x 3 x 3 x 3 = 4536

so n = 4536 divided by 168 = 27

and 168n then equals 4536 which is 14 x 324

168n=k324
324/168=n/k
81/42=n/k
27/14=n/k
27=3^3
14=2x7
n=27 & k=14
God bless you.