Exercises: The set of natural numbers

Walking through your city center, you see a plaque on an ancient monument indicating its year of construction in Roman numerals: MDCCCXCVII. Convert that date into our decimal number system.

In the decimal number system, the value of a digit depends on its position. In the number 7 475, state the value represented by the first "7" (starting from the left) and the value represented by the second "7". How many times greater is the first than the second?

Lucía wants to arrange her collection of $36$ stickers into a perfect square on her album cover. How many stickers must she place along each side of the square so that none are left over? Draw a simple diagram of the solution.

You can do it on paper and send a photo with your answer.

The estimated population of a social network in 2023 reached $1,205,040,300$ active users.

1. a) Write how this number is read in words.
2. b) Give its polynomial expansion using powers of base $10$.
3. c) What is the place value of the digit $5$ in this number?

Consider the natural number $N = 47,082$.

1. a) If we swap the thousands digit with the tens digit, what is the difference between the original number and the new number obtained?
2. b) How many complete hundreds are there in total in the number $N$?
3. c) Write the natural number closest to $N$ that ends in three zeros.

In a logistics warehouse, shoe boxes are arranged to form perfect cubes. If a shelf has a square base of $7 \times 7$ boxes and they are stacked to reach the same height to form a cube:

1. a) How many boxes are there in total on the shelf? Express it as a power and calculate its value.
2. b) If $150$ boxes are sold, what is the maximum number of boxes that can still form a perfect square on the warehouse floor? How many boxes would be left over in that case?

The Pythagoreans studied polygonal numbers. Look at the following sequence of hexagonal numbers and answer:

1. Draw or describe what the fourth hexagonal number pattern ($H_4$) would look like.

2. How many dots will pattern $H_5$ have?

3. Knowing that the formula to calculate the hexagonal number in position $n$ is $H_n = 2n^2 - n$, work out the value for $H_{12}$.

You can do it on paper and send a photo with your answer.

In a mathematical logic challenge, you are given an expression that is missing parentheses for the result to be correct. Place the necessary parentheses so that the equality holds:

Once solved, explain step by step the order of operations you followed to arrive at the result of $23$.

A four-digit natural number meets the following conditions:

• It is a palindromic number (it reads the same forwards and backwards).

• The sum of its digits is $24$.

• The thousands digit is three times the hundreds digit.

Reasonably determine which number it is and represent it using its expanded form.