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.

@LostInParadise Oh . . . your post with “40 is greater than 09” sounded like rambling irrelevance to me, so I re-read it and your OP, and realized you were intending an entirely different question from what I and @Blackwater_Park assumed you meant. That is, it’s so easy to look at numbers and see whether they end in 00 or not, that my brain overlooked the possibility you meant that there was a book with numbers in it, and somewhy we supposedly care if one of them’s divisible by 100 or not, but either we’re not allowed to just look at the numbers, or we think glancing at 70 numbers scanning for 00 at the end is not quick enough for us.

Though, now that I know what the problem is (and no, I’ve never heard of THAT problem before), it seems entirely obvious that you’re talking about whether a block 70 wide overlaps a multiple of 100 or not, so yeah, you could just look at the top or bottom number and assess that.

But why would anyone ever make a book with 70 consecutive integers in it?

And even if someone did that, why would anyone know that property of the book, and want to know if any were divisible by 100?

On the other hand, I actually CAN imagine an actual real-world problem equivalent to this, in computer programming. Like you’ve got a bunch of records in a list, and you want to know whether you’ve got a particular record in a block, and you know they’re consecutive, but you only have the last one’s ID number.

But it also seems really obvious, since if you know how many (n), yeah you can just realize the range is from the first ID to ID + n, or from the last ID to last ID – n.