The least common multiple (LCM) of 8, 10, 16, and 20 is the smallest number that is divisible by each of these numbers. We can find the LCM by factoring each number into its prime factorization and then multiplying the highest powers of each prime that appears in any of the factorizations.
Factoring the numbers:
* 8 = [tex]2^3[/tex]
* 10 = [tex]2*5[/tex]
* 16 = [tex]2^4[/tex]
* 20 =[tex]2^2*5[/tex]
Finding the highest power of each prime:
* 2: We have [tex]2^4[/tex] from 16, so the highest power is 4.
* 5: The highest power of 5 is 1 (it appears only in 10 and 20).
LCM [tex]= 2^4 * 5 = 160[/tex]
Since we want a number that leaves a remainder of 6 when divided by 160, we can add 6 to the LCM.
Least number = LCM + 6 = 160 + 6 = 166
Therefore, 166 is the least number that leaves a remainder of 6 when divided by any of the numbers 8, 10, 16, and 20.