Exercises: Natural numbers

Work out the result of the following combined operation with natural numbers, respecting the order of operations:

$15 - 3 \cdot (4 + 2) + 2^3 \cdot 5$

In an integer division of natural numbers, the divisor is $14$, the quotient is $23$ and the remainder is $5$. Work out the value of the dividend.

Simplify the following expression using index laws and express the result as a single power:

$2^4 \cdot 5^4 \cdot 10^2$

Work out the exact value of the following operation with roots of natural numbers:

$\sqrt{144} + \sqrt[3]{125} - \sqrt[4]{16}$

A restaurant offers a set menu consisting of $3$ first courses, $4$ main courses and $2$ different desserts. How many different menu combinations (first course, main course and dessert) can be chosen?

In a 4th-year ESO class, Lucía tries to quickly calculate the result of the operation $2^4 \cdot 5^4$ to find out how many combinations a padlock has. She writes in her notebook:

$2^4 \cdot 5^4 = (2 \cdot 5)^{4+4} = 10^8 = 100\,000\,000$.

Lucía argues: "Since it is a multiplication of powers, you have to multiply the bases and you must always add the exponents".

1. a) Find and clearly explain the conceptual reasoning error made by Lucía, indicating which two different properties of powers she has incorrectly mixed.
2. b) Correct the calculation by applying the appropriate property for powers with the same exponent ($a^m \cdot b^m = (a \cdot b)^m$) and find the exact final result without needing to calculate $2^4$ or $5^4$ separately.

A secondary school in Sevilla organizes an end-of-year field trip to Doñana National Park for $146$ 4th-year ESO students and accompanying teachers. For the transfer, they hire minibuses from an Andalusian company with $24$ seats each.

1. a) Model this situation using the fundamental relationship of Euclidean division ($D = d \cdot c + r$), identifying which real-life quantity in the problem is represented by the dividend ($D$), the divisor ($d$), the quotient ($c$), and the remainder ($r$).
2. b) Perform the Euclidean division and interpret how many minibuses will travel completely full and how many passengers will occupy the minibus that is not full.
3. c) Reason how many minibuses in total the school must hire so that no one is left behind, and calculate how many empty seats will remain in the last vehicle.

Halley's Comet orbits the Sun and is visible from Earth approximately every $76$ years. Its last recorded appearance was in the year $1986$. Will it be visible in the year $3050$? If not, what will be the closest year to $3050$ in which it can be observed?

Use prime factorisation and index laws to fully simplify the following expression, giving the result as a natural number:

$\frac{18^3 \cdot 12^4}{24^2 \cdot 9^4}$

A farmer wants to plant $1000$ apple trees forming a perfect grid (a square with the same number of trees in each row and in each column). How many trees will make up the side of the largest possible square? How many apple trees will be left unplanted?

In an integer division between natural numbers, the divisor is $d = 18$ and it is known that the remainder obtained is exactly three times the quotient ($r = 3 \cdot c$). The quotient is a non-zero natural number ($c \ge 1$).

1. a) Taking into account the mandatory condition of Euclidean division that requires $0 \le r < d$, determine with reasoning all the possible values that the quotient $c$ can take.
2. b) For each of the valid values of $c$, reconstruct and calculate the corresponding dividend $D$ using the relationship $D = d \cdot c + r$.
3. c) A classmate claims that if the dividend were $D = 126$, the remainder would be $r = 18$. Explain why that claim is mathematically impossible in an integer division according to the properties of the remainder.

An agricultural cooperative in Gandía (Valencia) prepares square boxes of selected oranges for export. Each layer of the box contains an $n \times n$ grid of fruits forming a perfect square. The manager has a batch of $500$ freshly harvested oranges and wants to form the largest possible square layer. Before doing the math, he estimates by eye that the side of the grid will have between $30$ and $35$ oranges.

1. a) Reasonably estimate between which two consecutive whole tens the number of oranges per side must lie, knowing that $20^2 = 400$ and $30^2 = 900$. Based on this, assess whether the manager's forecast is reasonable or overestimated.
2. b) Check by calculating the squares of the relevant natural numbers what the largest perfect square less than or equal to $500$ is, deducing how many oranges per side the box will have.
3. c) Calculate how many oranges are left over from the batch of $500$ and how many would be needed to complete the next perfect square. Check that the sum of both quantities matches the theoretical difference between two consecutive squares ($2n + 1$).

Observe the pattern of the powers of $7$ ($7^1=7$, $7^2=49$, $7^3=343$, $7^4=2401$, $7^5=16807$, ...). Use the pattern followed by the units digits to determine what the last digit of the number $7^{2025}$ will be.

A square mosaic is formed by square tiles of $1 \text{ cm} \times 1 \text{ cm}$. The artist decides to enlarge the mosaic by adding one more row and one more column of tiles to the edges to form a larger square. If $41$ new tiles were needed for this extension, how many tiles made up the original mosaic?

To calculate the numerical value of the expression $\frac{6^5 \cdot 5^5}{15^4 \cdot 2^4}$, three 4th-year ESO students propose different strategies:

• Aitor proposes calculating the expanded value of each power ($6^5 = 7776$, $5^5 = 3125$, etc.), multiplying the numerators and the denominators, and finally dividing the two resulting numbers.

• Bea proposes decomposing the bases into prime factors ($6 = 2 \cdot 3$ and $15 = 3 \cdot 5$), applying the properties of powers with the same base to group the factors $2$, $3$, and $5$, and simplifying by subtracting exponents.

• Carlos proposes noting that in the numerator both powers share the exponent $5$ and in the denominator both share the exponent $4$, so he groups the bases: $(6 \cdot 5)^5$ in the numerator and $(15 \cdot 2)^4$ in the denominator.

1. a) Carry out step by step the complete solution following Bea's method and following Carlos's method until obtaining the final natural result.
2. b) Compare the three methods, arguing the advantages and disadvantages of each in terms of time, manual calculation difficulty, and likelihood of making errors.
3. c) Justify which is the most efficient method for this specific case and reason why Aitor's method would be completely unfeasible if the exponents were $50$ and $40$ instead of $5$ and $4$.

In Spain, the check letter of the National Identity Document (DNI) is obtained through an official mathematical algorithm of the Ministry of the Interior: the 8-digit natural number of the DNI ($D$) is taken and divided by $23$. The remainder of said Euclidean division ($r$) determines the assigned letter according to a fixed table of 23 official characters (where $0$ corresponds to T, $1$ to R, ..., and $22$ to E).

1. a) Based on the fundamental relationship of Euclidean division ($D = 23 \cdot c + r$), justify mathematically why the official table must have exactly 23 distinct letters and why it is impossible for a DNI to have a remainder equal to or greater than 23.
2. b) Two twin sisters from Zaragoza have consecutive DNI numbers: Sofía's is $D$ and Elena's is $D + 1$. If Sofía's DNI gives a remainder $r = 14$, deduce what the remainder of Elena's DNI will be and reason whether both will have the same letter or different letters.
3. c) Suppose another person's DNI number yields a remainder of exactly $r = 22$. If we consider the person with the immediately following number ($D + 1$), deduce what their remainder will be and explain in detail what happens to the quotient $c$. Relate this change to the concept of "forming a new complete group" explained in the class notes.