Problems involving consecutive integers are often the most intuitive and the most fun. Let’s practice with a few.

• The product of three consecutive integers is 60. What is the smallest of those integers?

To solve this problem rigorously, we would need to set the smallest integer equal to x, the next smallest integer to x + 1, and the largest integer to x + 2. Then, we would need to multiply those three expressions and set the product equal to 60. Finally, we would need to solve for x, which is the smallest integer.

While this solution would work, it can be very time-consuming. Before trying a solution like this, I would recommend trying to find an easier, non-rigorous solution.

First, let’s try small numbers to get a bound of the possible solutions. If the smallest number was 5, the product of the three consecutive numbers would be 5 x 6 x 7. We know that 5 x 5 x 5 is 125, so 5 x 6 x 7 must be greater than 125. Since this product is larger than 60, we know the smallest number isn’t 5.

Now, let’s try 3. If 3 was the smallest number, the product would be 3 x 4 x 5. After some quick multiplication, we find that the product is 60, which is what we want. Thus, the answer is 3.

• What is the greatest whole number less than 1000 that:

(1) can be expressed as the sum of two consecutive whole numbers,

(2) can be expressed as the sum of three consecutive whole numbers, and

(3) can be expressed as the sum of five consecutive whole numbers? (Moems division M contest 2, 2017)

This problem is long, but it is very fun. Let’s attempt to solve it.

To be expressed as a sum of two consecutive whole numbers, the number must equal a + a + 1 for some a. This means that it must be a multiple of two, plus one. So, the number must be odd.

To be a sum of three consecutive integers, the number must equal b + b + 1 + b + 2. This means that it has to be a multiple of three.

To be expressed as the sum of five whole numbers, the number must equal c + c + 1 + c + 2 + c + 3 + c + 4, which equals 5c + 15. This means that the number must be a multiple of 5.

So, to satisfy all three conditions, the number must be a multiple of 3 and 5, so it must be a multiple of 15. Also, the number must be odd, as given by the first condition. We need to find the greatest integer that satisfies these conditions and is smaller than 1000.

Let’s try 995. This ends in a 5, so it is divisible by five. But the digits add up to 23, which is not a multiple of 3. What about 985? The digits in this number add up to 21, so this cannot be the answer. Now, let’s try 975. This is an odd multiple of five, and the digits add up to 21! That is a multiple of three. So, 975 is the answer.