Given a book containing 70 large consecutive numbers, what is a quick way of deteermining if one is of them is divisible by 100?

Quickest way is to ask you.

There will be two trailing zeros

All of them are divisible by 100. Some of them will have numbers to the right of the decimal point, others won’t.

Suppose that one of the numbers is divisible by 100. Its last two digits are 00. What can we say about how the last two digits of the last number compares to the last two digits of the first number?

This was asked (of, not by, students) and answered in 4th grade at my lower school.

I would think that the answer is a little, though not much, beyond 4th grqde. More like 6th grade. Do you recall it?

If the smallest number ends in >=30, then the answer is yes, otherwise, no.

@LostInParadise Of course I remember, as I said I did. Ends in 00, because of course it does. Multiply any number by 10, and the answer is you move the decimal place by 1, which for integers adds one zero to the end. It’s like one of the simplest and most basic pieces of math knowledge, and sort of a tell for whether someone older than 10 had difficulties with math class or not.

First introduction I remember at my (private, good) lower school was in 4th grade, when covering multiplication tables, and then it was reinforced many times in 5th grade.

We even learned it twice as often, because it was a selling point for the metric system.

And then we learned it again later when learning (or ensuring we still knew) what exponents were, and in each ensuing science class, whether astronomy, physics, chemistry, etc.

Imagine that the 10th to last number has the last two digits as 00. Then the last number would end in 09.

The number to the left of 00 has the last two digits of 99. Moving (70–10) = 60 numbers to the left has the first number ending in 100 – 60 = 40.

Note that 40 is greater than 09. In general, the last two digits of the last number will be smaller than the last two digits of the first number whenever 00 is the last two digits anywhere. I hope this makes sense intuitively. Obviously, when 00 is not covered, the last 2 digits of the last number are greater than the last two digits of the first number.

I asked this question to ChatGPt, Link. It did not find the answer, but it did a good job of explaining the answer after I gave it.