Some numbers have been listed below. Part a) Which of these numbers leaves a remainder of one when divided by 13?
One multiplied by 13 is equal to 13. We can calculate the next few numbers in the 13 times table by adding 13 repeatedly. Adding 13 to 13 gives us 26. Therefore, two multiplied by 13 is 26. Three multiplied by 13 is equal to 39. Four multiplied by 13 is equal to 52. And five multiplied by 13 is equal to 65.
We were asked to find a number that leaves a remainder of one when divided by 13. This means we need to find a number that is a multiple of 13 plus one or one more than one of the numbers in the 13 times table. We’ve already worked out that five multiplied by 13 is equal to 65. Adding one to this gives us 66.
This means that the number in the list that leaves a remainder of one when divided by 13 is 66. 66 divided by 13 is equal to five remainder one.
The second part of the question says the following. Part b) Write down all of the prime numbers in this list.
Any prime number has exactly two factors, one and itself. This means that all prime numbers are only divisible by one and the number itself. All of the even numbers in the list, 60, 62, 64, 66, and 68, are divisible by two. This is because every even number is a multiple of two.
65 can be divided by five as any number ending in five or zero is a multiple of five. 63 is not a prime number as it is divisible by three. 63 divided by three is equal to 21. The three remaining numbers 59, 61, and 67 are all prime as they only have two factors.
The only factors of 59 are one and 59; the only factors of 61 are one and 61; and in the same way the only factors of 67 are one and 67.
The final part of the question says the following. c) Which of these numbers could represent the volume of a cube, where the side length of the cube is an integer?
An integer is a whole number value. The lengths of every side of a cube are equal and its volume is calculated by multiplying 𝑥 by 𝑥 by 𝑥. This gives us 𝑥 cubed. This means that the only numbers that could represent the volume of a cube are the cube numbers.
In order to cube a number, we multiply it by itself and itself again. For example, two cubed is equal to two multiplied by two multiplied by two. This is equal to eight as two multiplied by two is equal to four and multiplying this by two gives us an answer of eight. Likewise, three cubed is equal to 27 as three multiplied by three is equal to nine and nine multiplied by three equals 27.
Four cubed is equal to four multiplied by four multiplied by four. Four multiplied by four is equal to 16 and 16 multiplied by four is equal to 64. This means that the number from the list that could represent the volume of a cube is 64.
The cube would have volume 64 units cubed and side length four units.
When answering this type of question, you don’t have to use every single one of the numbers in the list. Also, you could potentially use one of the numbers more than once, for example in part a) and part b).
In this case though, the answer to part a) is 66, part b) 59, 61, and 67, and part c) the answer is 64.