The World of Algorithms

What happens to the number of columns you need to add if you change a $3$-digit number addition to a $5$-digit number addition?

If counting dots for the sum $15 + 15$ takes $30$ seconds (at $1$ second per dot), and the column algorithm takes $4$ seconds per column, which method is faster?

If you have a group of $10$ tens, what does the addition algorithm instruct you to do in terms of carrying?

Suppose you are adding $456$ and $789$. After adding all the aligned columns, you find you have a 'carry' of $1$ remaining at the very end. What must you do according to the algorithm?

When adding $12,345$ and $678$ using the algorithm, which digit in $12,345$ is aligned with the $6$ in $678$?

Suppose we are adding two digits $d_1$ and $d_2$ in a column with an existing carry of $c = 1$. If the resulting sum in that column is $10$, what are the new values for the digit written below and the carry for the next column?

The digit-by-digit algorithm is described as 'much faster' than the dot method. If adding two 6-digit numbers takes $12$ basic operations using the algorithm, roughly how many dots would have to be drawn to find the sum of $500,000$ and $500,000$ using the explicit method?

A visual representation of the algorithm shows 'Carry' as a small marble moving to the next column. In the addition of $7,432 + 2,568$, which specific marble movements occur?

Suppose a system uses 'Base-6' instead of 'Base-10'. The algorithm remains the same, but the carry is set to 1 if the sum is $\ge 6$. If we add the numbers $(45)_6$ and $(23)_6$, what are the resulting carry values generated for the second column and the final 'Step 5' overflow?

An algorithm researcher defines 'Efficiency' ($E$) as $E = \frac{V}{S}$, where $V$ is the numerical value of the sum and $S$ is the number of steps (digit-additions) in the column algorithm. Calculate the change in efficiency when adding $100+100$ compared to $900+900$.

When following the addition algorithm for the problem $642 + 258$, what is the specific value of the 'carry' that is set after calculating the sum in the tens column (the middle digits)?

Suppose you are using the step-by-step algorithm to add two numbers, $A$ and $B$. In Step 2, you calculate the sum of the rightmost digits to be $13$. According to the algorithm, which digit should be written directly below those two digits, and what value should the 'carry' be set to for the next step?

How many columns will be processed in the addition of $456 + 78$?

In the problem $12 + 8$, what is the value of carry when you move from the units column to the tens column?

During the execution of the algorithm for $76 + 48$, a trace of the 'sum' calculated at each column (including incoming carries) is recorded. What is the sequence of these sums from right to left?

Suppose we are adding two $3$-digit numbers using the Indian place-value system. If the result of the hundreds column addition (Step 3) is a sum of $13$ and the carry was $1$, what instruction must be executed to complete the final answer according to Step 5 of the algorithm?

When executing the addition algorithm for $555 + 446$, identify the step where the carry value is set to $0$.

Suppose we add $A$ and $B$, both are $n$-digit numbers. If the addition generates a carry in every possible column (from units to the leftmost column), what is the minimum possible value for the sum of the digits of the resulting $(n+1)$-digit sum?

An explorer finds an ancient tablet that describes an addition algorithm. It specifies that for $567 + 433$, the digits must be aligned by their rightmost ends. If the explorer follows the steps and reaches the final column (hundreds), what is the sum of digits plus the incoming carry for that column, and does it trigger Step 5?

During the addition of two 4-digit numbers, the algorithm finds that in the hundreds column (the third column from the right), the sum of the digits plus the carry is 19. What specific value is written in the result row for this column, and what is the carry value for the next column?

The subtraction algorithm is applied to find $\text{gcd}(13, 5)$. Following the rule $(m, n) \rightarrow (n, m-n)$ and reversing if $m < n$, which sequence of pairs is correct?

A table displays the trace of an algorithm to find divisors of $n = 16$ by checking $j$ from $1$ to $16$:

If this process continues until $j = 16$, how many elements will be in the final list-of-divisors?

In the context of algorithm efficiency, why is the range for checking common divisors of $m$ and $n$ restricted to $1$ to $\text{min}(m, n)$ instead of $1$ to $\text{max}(m, n)$?

If an algorithm finds that the set of common divisors for two numbers is $\{1, 2, 4, 8, 16\}$, what is the value of the Highest Common Factor (HCF)?

Suppose we have two lists of divisors, $L_m = [1, 2, 3, 4, 6, 8, 12, 24]$ and $L_n = [1, 2, 3, 5, 6, 10, 15, 30]$. If an algorithm creates a new list $L_{common}$ by taking elements present in both, what is the average (mean) of the elements in $L_{common}$?

Two gears $G1$ and $G2$ are engaged. $G1$ has 126 teeth and $G2$ has 162 teeth. An algorithm calculates the GCD of the tooth counts to determine the number of teeth that pass before the same two teeth meet again. What is the sum of the digits of this GCD?

An algorithm stores the 'list-of-divisors' for $n=60$ in an array $L$. If the algorithm is modified to only store divisors $j$ that satisfy the condition $j > \sqrt{60}$, how many elements will be in the list $L$?

A data structure stores the divisors of 144 in an ordered list $L_1$ and the divisors of 216 in an ordered list $L_2$. An algorithm creates a new list $L_3$ containing only the common divisors. What is the sum of the last three elements of $L_3$?

A farmer has 945 cows and 2475 sheep. He wants to form them into flocks, keeping cows and sheep separate and having the same number of animals in each flock. If these flocks are as large as possible, and the farmer charges a grazing fee of Rs 200 per flock, what is the total grazing fee he will collect?

A tailor has two pieces of cloth of widths 112 cm and 140 cm. She wants to cut both pieces into strips of equal width, as wide as possible, to make ribbons. After cutting, she finds the total number of ribbons. If each ribbon is 2 meters long, what is the total area of all the ribbons produced in square centimeters?

Consider the execution of the algorithm to find the divisors of $n = 42$. A student records the 'list-of-divisors' at various stages. Which number will be added to the list immediately after the number $7$ is added?

If the divisor algorithm is applied to the number $n = 12$, which of the following diagrams correctly represents the 'list-of-divisors' state immediately after the step where $j = 5$ is completed?

A student is following the algorithm to find the divisors of $n = 40$. While checking the numbers in the sequence $1, 2, 3, \dots, 40$, which of the following is the first value of $j$ that the algorithm will check but not add to the 'list-of-divisors'?

A student correctly follows the algorithm for $n = 4$. What is the size (number of elements) of the final 'list-of-divisors'?

During the execution of the divisor algorithm for $n = 22$, what is the result of the divisibility test and the action taken for $j = 11$?

If the algorithm for finding divisors is executed for $n = 49$, which of the following describes the sequence of $j$ values that result in an update to the 'list-of-divisors'?

An algorithm for finding divisors of $n = 18$ is being executed. Let $S$ be the sum of all elements currently in the 'list-of-divisors'. What is the value of $S$ immediately after the algorithm processes $j = 9$?

In the execution of the divisor algorithm for $n = 81$, let $A$ be the sum of all values of $j$ that failed the divisibility test and $B$ be the sum of all values of $j$ that passed the test. Calculate the value of $(A + B)$ and identify its relationship to $n$.

For a mystery number $n$, the algorithm's 'list-of-divisors' is $[1, 2, 4, 5, 10, 20]$. If the algorithm is currently at step $j = 25$, and the list has not been updated since $j = 20$, what is the smallest possible value for the final element that will eventually be added to this list?

During the execution of the algorithm for $n = 36$, a student calculates the product of the last two elements currently in the 'list-of-divisors' every time an update occurs (starting from when the list has at least 2 elements). What is the maximum value this product reaches throughout the execution?

Suppose you are comparing two lists of divisors to find common elements. For any two positive integers $a$ and $b$, which number is guaranteed to be the first element added to the list of common divisors?

When computing the GCD of two large numbers using the systematic list-comparison method, we first obtain two lists of divisors sorted in increasing order. What is the standard next step to determine the GCD?

A student is finding the Greatest Common Divisor (GCD) of $m = 48$ and $n = 72$ by listing their divisors. If the list of common divisors is $[1, 2, 3, 4, 6, 8, 12, 24]$, which element of this list represents the GCD?

If we compare $divisors(25) = [1, 5, 25]$ and $divisors(10) = [1, 2, 5, 10]$, what is the largest number that appears in both lists?

If an algorithm to find the divisors of $n$ takes $T$ seconds for a $3$-digit number, it is observed that for a $6$-digit number, the same algorithm takes approximately $1000T$ seconds. This suggests the work is proportional to which of the following?

An algorithm finds the common divisors of two numbers $m$ and $n$ and stores them in a list $L$. If the sum of all elements in $L$ is $12$ and the GCD of $m$ and $n$ is $6$, which of the following could be the complete list $L$?

When executing the algorithm for $\text{gcd}(m, n)$ by comparing sorted divisor lists, the complexity (amount of work) is related to the number of divisors. If $m=2^5$ and $n=3^5$, how many elements will be in the 'common-divisors' list?

An algorithm is written to find the GCD of $m = 10!$ and $n = 11!$. By expressing the numbers as products, the algorithm identifies common factors. What is the value of the GCD, and how many distinct prime factors does it have?

Three alarm clocks are set to beep at intervals of $105$ minutes, $140$ minutes, and $175$ minutes. To find the largest possible interval $X$ that divides all three, a student uses the divisor-list comparison method. What is the difference between the largest common divisor and the second-largest common divisor found in the list?

A school organizes its students into groups for a parade. There are $315$ students in the primary section and $441$ students in the middle section. Each group must have the same number of students and must contain students from only one section. If the groups are made as large as possible, what is the total number of groups formed from both sections combined?

Why is it necessary to find the 'rightmost element' of the 'common-divisors' list to identify the greatest common divisor in this algorithm?

In the First Algorithm to find the $gcd$ of two numbers, we check if each element $x$ from 'divisors-of-m' exists in 'divisors-of-n'. If $m = 12$ and $n = 15$, which of the following divisors of $12$ will fail this check and NOT be added to the 'common-divisors' list?

If the First Algorithm is used to find $gcd(16, 24)$, and the final 'common-divisors' list is identified as $[1, 2, 4, 8]$, which specific element is reported as the final answer in Step 5?

According to the systematic steps of the First Algorithm, which list must be completely generated before the comparison process in Step 4 can begin?

If $m$ is a prime number $p$, and $n$ is another prime number $q$ (where $p \neq q$), what will be the length of the 'common-divisors' list generated by the algorithm?

In the list $[1, 2, 5, 10]$, which represents the common divisors of $20$ and $50$, the index of the GCD (using 1-based indexing) is:

Why does the First Algorithm for gcd check numbers from $1$ up to $m$ to find 'divisors-of-m'?

A student is tracing the First Algorithm for $gcd(480, 720)$. After generating divisors-of-480 and divisors-of-720, the comparison step begins. Which of the following numbers in divisors-of-480 is the smallest value that is greater than 10 and is NOT included in the common-divisors list?

In the First Algorithm for $gcd(m, n)$, if $m = 1155$ and $n = 15015$, the algorithm determines the common-divisors list. What is the arithmetic mean (average) of the smallest and the largest elements in the common-divisors list?

During the execution of the First Algorithm for $gcd(288, 432)$, the list divisors-of-288 is generated. Let $D_{288}$ be this list. A student labels the elements of $D_{288}$ as $d_1, d_2, \dots, d_{18}$. How many of these elements will be successfully added to the common-divisors list in Step 4?

In a data structure representing a sorted list of common divisors $[1, 2, 4, 8]$, which position (or index) holds the Greatest Common Divisor (GCD)?

Consider an algorithm that tracks the steps of a search. It uses a data structure called visited_nodes which starts as an empty list []. In each step, if it visits a new number, it adds it to the end of the list. After visiting the numbers $5, 12,$ and $7$ in that order, what is the state of visited_nodes?

An algorithm processes a list of integers L = [15, 22, 9, 31, 18] and creates a new data structure Even_List containing only the even numbers from L, maintaining their original relative order. What are the contents of Even_List?

When building an algorithm, why do we assign names like list_of_divisors or current_carry to intermediate quantities?

A student is writing an algorithm to compare two lists of divisors, $L1$ and $L2$, both sorted in ascending order. The goal is to find the 'second largest common divisor'. If $L1 = [1, 2, 3, 4, 6, 8, 12, 24]$ and $L2 = [1, 2, 3, 4, 6, 9, 12, 18, 36]$, what value will the algorithm identify?

An algorithm uses a data structure named exponent_map to store the prime factorization of a number $N$ as a list of exponents for the primes $[2, 3, 5, 7, \dots]$ in order. For example, $12 = 2^2 \times 3^1 \times 5^0$, so its map is $[2, 1, 0]$. If the algorithm computes the exponent_map for $N = 70$, what is the sum of all elements in that list (up to the largest prime factor)?

In a certain algorithm, a 'difference-list' is created from a sorted list of primes $P = [2, 3, 5, 7, 11, 13]$. The $i^{th}$ element of the difference-list is calculated as $P[i+1] - P[i]$. Which of the following represents the correctly generated 'difference-list'?

An algorithm Prime-Power-Organize represents a number $N$ as a nested list of pairs [[p1, e1], [p2, e2], ...] where $p$ is a prime factor and $e$ is its exponent. If the algorithm processes $N = 504$, it stores this in a data structure $D$. It then performs a transformation to create a new list $E = [e1, e2, ...]$ containing only the exponents. What is the sum of the elements in the data structure $E$?

Consider an algorithm that tracks the 'Density' of divisors. For a number $n$, it creates a data structure Div-List. It then calculates the 'Gap-List' where $G[i] = Div-List[i+1] - Div-List[i]$ for all elements in the sorted list. If $n=24$, what is the maximum value found in the resulting Gap-List data structure?

In a logistics algorithm, a data structure Package-Weights stores weights in kg: $[15, 22, 18, 25, 30]$. To optimize shipping, the algorithm creates a list Cumulative-Load where the $n$-th element is the sum of the first $n$ weights. It then checks a constraint: any load exceeding 60 kg requires an extra vehicle. The algorithm flags the index in Cumulative-Load where this first happens. What is the value of the flagged index (using 1-based indexing) and the value stored at that index?

Consider an algorithm that scans for the greatest common divisor of $m=18$ and $n=27$ by checking $k$ from $1$ to $\min(m, n)$. For which of the following values of $k$ will the 'is-common-divisor' check be true and cause an update to the most-recent-common-divisor variable?

In the optimized algorithm for GCD, we maintain only the most-recent-common-divisor instead of a full list of common divisors. Which statement best explains why earlier common divisors like $2$ can be safely overwritten when a later common divisor like $6$ is found?

An algorithm is designed to find common divisors of two numbers $x$ and $y$. If the algorithm checks every integer from $1$ to $x$ and then every integer from $1$ to $y$ separately, it performs $x + y$ checks. If we optimize this to a single scan up to $\min(x, y)$, how many checks are saved when $x = 150$ and $y = 400$?

When using the 'most-recent-common-divisor' variable approach for $m = 12$ and $n = 16$, the variable is updated every time a common divisor is found while scanning from $1$ to $\min(m, n)$. What is the third value stored in this variable during the execution?

Suppose we execute a 'most-recent-common-divisor' algorithm for $gcd(60, 90)$. The algorithm initializes the variable to $1$ and scans $k$ from $2$ to $60$. What is the value of the variable immediately after the loop finishes checking $k = 10$, but before it checks $k = 11$?

When improving a divisor-finding algorithm for $gcd(m, n)$ to 'do away with lists', we maintain only the most-recent-common-divisor. In a trace where $m = 30$ and $n = 45$, the variable is updated at $k = 3$, $k = 5$, and finally $k = 15$. Why are the values $3$ and $5$ considered 'useless' once the loop reaches $k = 15$?

Consider the following algorithm for finding the $gcd$ of two numbers $P$ and $Q$.

If $P = 48$ and $Q = 18$, identify the set of values that $j$ takes in Step 3 which specifically result in an update to the variable $G$.

An algorithm analyzer is looking at the work growth of the GCD scan. If the minimum of two numbers increases from a $5$-digit integer to a $9$-digit integer, by what factor does the worst-case number of iterations increase in the 'combined scan to $\min(m, n)$' algorithm?

A student proposes a 'Step-Skip' improvement: for $gcd(m, n)$, scan $k$ from $1$ to $\min(m, n)$, but only check even numbers if both $m$ and $n$ are even. If $m = 100$ and $n = 250$, how many iterations are performed using this 'Step-Skip' scan compared to a standard $1$ to $\min(m, n)$ scan?

Two algorithms are tested on $m=120$ and $n=180$. Algorithm 1 finds all common divisors and stores them in a list $L$. Algorithm 2 uses the 'most-recent' variable method. If we define 'efficiency loss' as the number of elements in $L$ that are smaller than the actual GCD, what is the 'efficiency loss' for these specific inputs?

The table below compares the amount of work done by two different algorithms, $A$ and $B$, for numbers with different values. Based on the growth shown, which statement accurately describes the workload of Algorithm $B$?

A specific algorithm for processing a number takes $5$ units of work for every digit in the number. If we replace a $2$-digit number with a $10$-digit number, how many times does the total workload increase?

Which mathematical operation is used in the 'reduction step' of Āryabhaṭa’s improved GCD algorithm to make it more efficient than Euclid's subtraction?

Using the column addition algorithm for two 6-digit numbers, what is the minimum number of single-digit additions performed?

When we analyze the 'list-of-divisors' algorithm that checks every $j$ from $1$ to $n$, we find it is inefficient for large $n$. If a computer can perform $10^7$ checks per second, what is the smallest number of digits a number $n$ can have such that the algorithm might take more than 10 seconds to complete?

In the analysis of addition algorithms, counting dots for $50 + 50$ takes 100 units of work. Column addition for $50 + 50$ takes roughly 2 units of work (adding two columns). What is the ratio of work (Counting : Column) for the sum $10^4 + 10^4$?

If the effort to execute an algorithm on a number $n$ is proportional to the number of its divisors, which of the following inputs would likely require the most work?

An algorithm's work $W$ is related to the number of digits $d$ by the formula $W = k \cdot 10^d$ for some constant $k$. If increasing the number of digits from $3$ to $4$ adds $9,000$ units of work, what is the value of the constant $k$?

A student claims that the column-addition algorithm is 'digit-proportional'. If adding two $5$-digit numbers takes $T$ milliseconds, approximately how much time would it take to add two $500$-digit numbers using the same logic?

A programmer is deciding between two GCD algorithms for a mobile app. Algorithm 1 scans up to $min(m, n)$. Algorithm 2 uses repeated subtraction. If the typical inputs are $m=10^9$ and $n=3$, which algorithm will be more likely to cause the app to freeze, and why?

A student is finding the greatest common divisor of two numbers $m$ and $n$. If $m < n$, what is the first step they should perform to follow the standard Euclid-style algorithm steps precisely?

In the subtraction-based algorithm for $\text{gcd}(m, n)$, if the algorithm starts with $m = 21$ and $n = 21$, which of the following is the immediate next reduction state according to the step 'Otherwise, reduce the problem to compute $\text{gcd}(n, m - n)$'?

When comparing two algorithms for finding the greatest common divisor, if the number of steps in Algorithm A is proportional to the values of the numbers, and the number of steps in Algorithm B is proportional to the number of digits in the numbers, which statement is true for very large inputs?

If we use the improved division-based algorithm to find the greatest common divisor of $80$ and $12$, what is the result of the first reduction step $\text{gcd}(n, m \pmod n)$?