In order to find all of the numbers that are multiples of both 8 and 18, we need to find the least common mutliple (LCM) of 8 and 18. The easiest way to do this would be to list out the multiples of 8 and 18 and determine the smallest one that is common to both. 

First, let's list the first several multiples of eight:

8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88 . . .

Next, we list the first several multiples of eighteen:

18, 36, 54, 72, 90, 108, 126, 144 . . .

By comparing the multiples of eight and eighteen, we can see that the smallest one that they share is 72. Thus, the LCM of 8 and 18 is 72.

Because the LCM is 72, this means that every multiple of 72 is also a multiple of both 8 and 18. So, in order to find all of the multiples less than ten thousand that are both multiples of 8 and 18, we simply need to find how many multiples of 72 are less than 10000, and to do this, all we have to do is to divide 10000 by 72.

When we divide 10000 by 72, we get 138 with a remainder of 64; therefore, 72 will go into ten thousand 138 times before it exceeds ten thousand. In other words, there are 138 numbers less than 10000 that are multiples of 72 and, by extension, also multiples of both 8 and 18. 

The answer is 138.

$\displaystyle 8,5$ are different kind of numbers. We have a composite number and a prime number, respectively. They share a factor of $\displaystyle 1$. Therefore, we just multiply both numbers to get an answer of $\displaystyle 40$.

Least common multiple is the smallest number that is divisible by two or more factors. Since $\displaystyle 2, 3$ are prime numbers and can't be broken down to smaller factors, we just multiply them to get $\displaystyle 6$ as our answer. 

We are inclined to multiply the numbers out however, if we divide both numbers by $\displaystyle 4$, we get $\displaystyle 1, 3$ remaining. These numbers are unit and prime numbers respectively and only share a factor of $\displaystyle 1$. To determine the least common multiple, we multiply the factor with the numbers remaining. Our answer is just $\displaystyle 4*1*3$ or $\displaystyle 12$. 

Both $\displaystyle 6, 8$ are even so we can divide both numbers by $\displaystyle 2$ to get $\displaystyle 3, 4$. We have a prime number and a composite number, respectively. They share a factor of $\displaystyle 1$. To determine the least common multiple, we multiply the factor with the numbers remaining. Our answer is just $\displaystyle 2*3*4$ or $\displaystyle 24$. 

Both numbers are divisible by $\displaystyle 6$, so the remaining numbers are $\displaystyle 3, 4$. We have a prime number and a composite number respectively. They share a factor of $\displaystyle 1$. To determine the least common multiple, we multiply the factor with the numbers remaining. Our answer is just $\displaystyle 6*3*4$ or $\displaystyle 72$. 

If we divide $\displaystyle 5$ for both  numbers, we get $\displaystyle 5, 25$. We do it the second time and we get $\displaystyle 1, 5$. Now we have a unit and a prime number. So we just multiply the factors and the remaining numbers to get $\displaystyle 5*5*1*5$ or $\displaystyle 125$. 

Both numbers are divisible by $\displaystyle 9$ because the sum of the digits are divisible by $\displaystyle 9$. We get $\displaystyle 4, 6$ as the remaining numbers. We can divide by $\displaystyle 2$ to get $\displaystyle 2, 3$. We have two prime numbers. Now, we multiply the factors and the remaining numbers to get $\displaystyle 9*2*2*3$ or $\displaystyle 108$.

Both $\displaystyle 26, 8$ are even so we can divide both numbers by $\displaystyle 2$ to get $\displaystyle 13, 4$. We have a prime number and a composite number respectively. They share a factor of $\displaystyle 1$. To determine the least common multiple, we multiply the factor with the numbers remaining. Our answer is just $\displaystyle 2*13*4$ or $\displaystyle 104$. 

We need to ensure that all the numbers share a common factor of $\displaystyle 1$. $\displaystyle 2, 4$ are even so let's divide those by $\displaystyle 2$. We get $\displaystyle 1, 2$ leftover along with the $\displaystyle 3$ that doesn't divide evenly with $\displaystyle 2$. Now that all these numbers share a common factor of $\displaystyle 1$, we multiply them all out including the $\displaystyle 2$ we divided out. We get $\displaystyle 2*1*2*3$ or $\displaystyle 12$.