𝑒 is an even number and 𝑠 is a square number. Part a) Give an example where 𝑒 plus 𝑠 is a prime number. Part b) Give an example where 𝑒𝑠 or 𝑒 multiplied by 𝑠 is a multiple of three.
As 𝑒 is an even number, it could be two, four, six, eight, 10, and so on. 𝑠 is a square number. The first five square numbers are one, four, nine, 16, and 25. To square any number, we multiply it by itself. One squared or one multiplied by one is one. Two squared is four. Three squared is nine. Four squared is 16. And five squared is 25, as five multiplied by five equals 25.
We were asked in part a) to give an example where 𝑒 plus 𝑠 is a prime number. A prime number has exactly two factors: the number one and itself. The first five prime numbers are two, three, five, seven, and 11. The only factors of two are one and two. The only factors of three are one and three. The only factors of five are one and five, and so on.
We need to find an example where 𝑒 plus 𝑠 is equal to one of these prime numbers. There are lots of possible answers here, one of which is when 𝑒 equals six and 𝑠 equals one. Six plus one is equal to seven. Therefore, 𝑒 plus 𝑠 is equal to seven, which is a prime number.
Other possibilities from our list would be 𝑒 equals two and 𝑠 equals one. They have a sum of three, which is a prime number. Likewise, 𝑒 equals four and 𝑠 equals one, this has a sum of five. Finally, 𝑒 equals 10 and 𝑠 equals one has a sum of 11. There are numerous other combinations that we could’ve chosen.
The second part of this question wants us to give an example where 𝑒𝑠 is a multiple of three. Let’s first rewrite our values for 𝑒 and 𝑠. 𝑒 was the even numbers and 𝑠 was the square numbers. The multiples of three are the numbers in the three times table: three, six, nine, 12, 15, and so on. We need to find an example where 𝑒𝑠 or 𝑒 multiplied by 𝑠 is one of these multiples.
Once again, there are lots of different possibilities. One example is when 𝑒 equals six and 𝑠 equals four. Six multiplied by four is equal to 24. Therefore, when 𝑒 is equal to six and 𝑠 is equal to four, 𝑒𝑠 is equal to 24, which is a multiple of three. In this case, since six is a multiple of three, we could actually pick any 𝑠-value and multiply it by six. And we would get an answer that is a multiple of three. Six multiplied by one, six multiplied by four, six multiplied by nine, and so on would all be multiples of three.