Which of the following is a multiple of both seven and two and of their product 14? Four, 21, 23, 84, or 91.
In this problem, we’re looking for a number that’s a multiple of three different numbers. It has to be a multiple of seven, also of two, and of 14, which we’re reminded is the product of seven and two. We’re given five possible answers. Now, one way to find the answer might be to list all the multiples of seven — seven, 14, 21, and so on — then all the multiples of two — two, four, six, and so on — then all the multiples of 14 — 14, 28, 42, and so on. Keep going until we find a number that is a multiple of all three numbers. But there’s actually a quicker way to find the answer here. And we can use reasoning to help.
Firstly, we can see that one of our numbers is two. What can we say about every single number that’s a multiple of two? We know that if a number is a multiple of two, it’s an even number. This is how we know what an even number is; it’s always a multiple of two. So if our answer is a multiple of two, we need to cross out all the odd numbers because we know the answer is going to be an even number. So we can cross out 21, we can cross out 23, and we can also cross out 91. This only leaves us with two possible answers. Is the answer likely to be four or 84? Again, we can use reasoning to help here.
If we look at our two possible answers, we can see that one of them is fairly small, the number four. We know that four is in the two times table because two times two equals four. But could we say it’s in the seven times table or the 14 times table? No, it’s too small. It’s impossible for the number four to be a multiple of seven and 14. And so without doing any calculations, we’ve narrowed down our answer to 84. Let’s just check whether we’re correct? How many twos are in 84? Well, to answer this question, we need to think about what we might double to get 84. 42 doubled is 84. And so we can say that 84 is definitely a multiple of two. We knew that anyway because it’s an even number.
But is it the multiple of seven? How many sevens are in 84? We know 10 sevens are 70, 11 sevens are 77, and so 12 sevens must be equal to 84. It’s a multiple of seven. And finally, let’s check whether it’s a multiple of the product of seven and two, which is 14. How many 14s are in 84? We’re going to have to count in 14s here, but there is a quick way to do this. 14 is double seven. So if we list all the multiples of seven, starting with seven and going all the way up to 84, then we can use this to help us find the answer.
Remember, every two lots of seven is a multiple of 14. So we just have to read every other number. 14, 28, 42 — these are all multiples of 14 — 56, 70, and 84. They’re all the even numbers in the seven times table. There are six lots of 14 in 84. We’ve just used our knowledge of times tables to prove the answer. But remember, we found the answer simply using reasoning. Any multiple of two is an even number. And unless we count zero, any multiple of 14 is gonna have to be 14 or more. The only possible answer we have is 84. So the multiple of seven, two, and their product 14 is 84.