CAT Number System, HCF & LCM Questions, LR & DI | Day 6 Preparation for CAT 2026

1. CAT Quantitative Aptitude: HCF and LCM

The Highest Common Factor (HCF) and Least Common Multiple (LCM) are important concepts in the Number System. Questions may involve prime factorisation, remainders, co-prime numbers, fractions or situations in which multiple events occur at regular intervals.

What are HCF and LCM?

• HCF: The greatest positive integer that divides two or more numbers exactly.

• LCM: The smallest positive integer that is divisible by two or more numbers exactly.

For example, the HCF of 36 and 48 is 12, while their LCM is 144.

Methods to calculate HCF and LCM

1. Prime factorisation method

Express each number as a product of prime factors.

For example:

$27 = 3^3,\quad 36 = 2^2 \times 3^2$

- HCF: Take the lowest powers of the common prime factors.

$ \text = 3^2 = 9$

- LCM: Take the highest powers of all prime factors.

$\text = 2^2 \times 3^3 = 108$

2. Division method for HCF

Divide the larger number by the smaller number. Continue dividing the previous divisor by the remainder until the remainder becomes zero. The last non-zero remainder is the HCF.

For example:

$48 = 18 \times 2 + 12$

$18 = 12 \times 1 + 6$

$12 = 6 \times 2 + 0$

Therefore, the HCF of 48 and 18 is $6$.

Important HCF and LCM formulas

For two positive integers (a) and (b):

$\text{HCF}(a,b)\times\text{LCM}(a,b)=a\times b$

For fractions in their simplest forms, the commonly used rules are:

$\text{HCF}=\frac{\text{HCF of numerators}}{\text{LCM of denominators}}$

$\text{LCM}=\frac{\text{LCM of numerators}}{\text{HCF of denominators}}$

CAT practice question 1: Find the number of integer pairs

How many unordered pairs of positive integers $(x,y)$ exist such that the HCF of $x$ and $y$ is $18$ and their product is $1,620$?

1. $2$

2. $8$

3. $1$

4. $3$

Solution:

Since the HCF is $18$, let

$x = 18a,\quad y = 18b$

where $a$ and $b$ are co-prime positive integers.

Given that $xy = 1620$:

$18a \times 18b = 1620$

$324ab = 1620$

$ab = \dfrac1620324 = 5$

The only co-prime positive integer factor pair of $5$ is $(1,5)$. Therefore, the unordered pair is

$x = 18 \times 1 = 18$

$y = 18 \times 5 = 90$

There is only one unordered pair.

Answer: Option 3 ($1$)

Note: If ordered pairs are counted separately, $(18,90)$ and $(90,18)$ give two pairs.

CAT practice question 2: Find when two lights flash together

A red light flashes three times per minute at regular intervals, while a green light flashes five times in two minutes at regular intervals. If both lights start flashing together, how many times do they flash together during the next hour, excluding the initial simultaneous flash?

Solution:

The red light flashes once every:

$\dfrac603 = 20$ seconds

The green light flashes once every:

$\dfrac1205 = 24$ seconds

They flash together at intervals equal to the LCM of $20$ and $24$:

$\text(20,24) = 120$ seconds

Since $120$ seconds equals $2$ minutes, they flash together every two minutes.

The number of subsequent coincidences in $60$ minutes is:

$\dfrac602 = 30$

Answer: $30$ times, excluding the initial simultaneous flash.

CAT preparation tips for HCF and LCM

• Use prime factorisation: It is often the quickest method when the numbers are easy to factor.

• Apply the product rule: If the HCF and one number are known, the LCM or the other number may be calculated directly.

• Recognise co-prime numbers: Their HCF is 1, so their LCM equals their product.

• Practise remainder-based problems: Identify whether the question asks for the greatest divisor or the smallest suitable multiple.

• Revise Number System fundamentals: Practise factors, multiples, prime numbers and co-prime numbers before attempting advanced questions.

2. CAT DILR: Data Interpretation Based on Radar Graphs

A radar graph, also called a spider chart, displays multiple variables on axes that extend from a common centre. Each axis represents a different parameter, and the plotted values can be compared across categories.

In CAT DILR, radar graph questions may test comparisons, averages, medians, rankings and the interpretation of multiple data points.

How to read a radar graph

• Each spoke represents one variable or evaluation parameter.

• The distance from the centre indicates the value for that variable.

• A larger value appears farther from the centre, provided the same scale is used.

• Different lines or shapes represent different categories, such as vendors or products.

For example, a vendor evaluation chart might compare Cost, Customer Service, Features, Quality, Reach and Reliability on a scale from 0 to 100.

Important calculations for radar graph questions

CAT practice question 1: Identify the parameter with the lowest median

Five vendors are evaluated on six aspects: Cost, Customer Service, Features, Quality, Reach and Reliability. Each aspect is scored from 0 to 100.

On which aspect is the median score the lowest?

1. Customer Service

2. Cost

3. Reliability

4. Quality

Solution:

Compare the median scores across all six parameters. Customer Service has a median of 50, which is lower than the reported median for every other parameter.

Answer: Option 1 (Customer Service)

CAT practice question 2: Comparing the Performance of Four Companies

A radar graph compares four companies across five parameters: Innovation, Product Quality, Market Reach, Customer Service, and Cost Efficiency. Each parameter is scored out of 100. The scores are given below.

Question: A company is shortlisted only if its average score across all five parameters is at least 72 and its scores in both Product Quality and Customer Service are at least 70. Which company or companies are shortlisted?

1. Company A only

2. Companies A and B only

3. Companies A and D only

4. Companies A, B and D only

Solution

The average score is calculated as:

$\text = \dfrac}5$

- Company A: $\dfrac3705 = \dfrac3705 = 74$

Product Quality $= 75$ and Customer Service $= 85$. Qualifies.

- Company B: $\dfrac3605 = \dfrac3605 = 72$

Product Quality $= 85$, but Customer Service $= 60$. Does not qualify.

- Company C: $\dfrac3555 = \dfrac3555 = 71$

The average is below $72$. Does not qualify.

- Company D: $\dfrac3705 = \dfrac3705 = 74$

Product Quality $= 80$ and Customer Service $= 75$. Qualifies.

Correct Answer: Option 3 - Companies A and D only.

CAT practice question 3: Evaluating the Performance of Four Retail Stores

A retail chain evaluates four stores using a radar graph with five parameters: Sales Growth, Customer Retention, Inventory Efficiency, Online Reach, and Profit Margin. Each parameter is scored out of 100.

The scores are given below.

A store is eligible for a performance award only if its Inventory Efficiency score is at least 70 and its Online Reach score is at least 70. Among eligible stores, the final performance index is the average of Sales Growth, Customer Retention, and Profit Margin.

Which eligible store has the highest final performance index?

1. Store P

2. Store Q

3. Store R

4. Store S

Solution

Step 1: Identify eligible stores.

A store must score at least 70 in both Inventory Efficiency and Online Reach.

• Store P: Inventory Efficiency = 60. Not eligible.

• Store Q: Inventory Efficiency = 75 and Online Reach = 75. Eligible.

• Store R: Inventory Efficiency = 80 and Online Reach = 85. Eligible.

• Store S: Inventory Efficiency = 65 and Online Reach = 60. Not eligible.

Only Stores Q and R qualify.

Step 2: Calculate their final performance indices.

The formula is:

$\text = \dfrac + \text + \text}3$

For Store Q:

$\dfrac2353 = \dfrac2353 \approx 78.33$

For Store R:

$\dfrac2253 = \dfrac2253 = 75$

Step 3: Compare the results.

Store Q scores approximately $78.33$, while Store R scores $75$. Therefore, Store Q has the higher final performance index among eligible stores.

Correct Answer: Option 2 - Store Q.

How to prepare for radar graph questions in CAT

• Practise visual comparisons: Learn to compare values without repeatedly rereading the entire chart.

• Revise averages and medians: These calculations frequently appear in data interpretation sets.

• Check the scale: A difference in scale can affect how a graph should be interpreted.

• Avoid assumptions: Use the plotted values rather than estimating from the overall shape when precise numbers are provided.

• Practise timed sets: Develop a habit of selecting the most efficient calculation method.

Useful prerequisites include percentages, simplification, approximation, mean, median, mode and observation-based reasoning.

3. CAT VARC: Articles and Their Uses

Articles are words used before nouns to indicate whether the noun refers to something specific or something nonspecific. The three English articles are a, an and the.

Correct article usage can help candidates improve grammatical accuracy and sentence understanding.

Types of articles

The choice between a and an depends on the sound, not simply the first letter of the word. For example, we say an honest person because the “h” is silent, but a university because the word begins with a “y” sound.

CAT practice questions on articles

Question 1: Who is _____ prime minister of the United Kingdom now?

Options:

1. The

2. A

Answer: Option 1 (The)

Explanation: The definite article “the” is used because the sentence refers to a specific officeholder.

Question 2: There is _____ nice restaurant near my house.

Options:

1. A

2. No article

Answer: Option 1 (A)

Explanation: “Restaurant” is a singular countable noun, and “nice” begins with a consonant sound. Therefore, a nice restaurant is correct.

Question 3: My grandparents live in _____ south of the city.

Options:

1. South

2. The south

Answer: Option 2 (The south)

Explanation: When referring to a specific geographical region or part of an area, the definite article is generally used.

Question 4: Is this _____ Australian breed?

Options:

1. A

2. An

Answer: Option 2 (An)

Explanation: “Australian” begins with a vowel sound, so an Australian breed is correct.

Common rules for using articles

• Use a and an with singular countable nouns when referring to them nonspecifically.

• Use the when the noun is specific, already identified or understood from context.

• Do not normally use articles before uncountable nouns when speaking about them in a general sense. For example, “Information is useful” is correct.

• Articles are generally omitted before languages, academic subjects and sports when referring to them in a general sense. However, articles may be necessary when the noun is specific or the sentence structure requires one.

• Do not place an article directly before a possessive determiner or most other determiners. For example, say “my book,” not “the my book.”

• Read regularly and practise sentence correction to understand article usage in context.

CAT VARC preparation tips for articles

• Revise noun types: Understand singular, plural, countable and uncountable nouns.

• Learn sound-based rules: Distinguish between vowel letters and vowel sounds.

• Practise sentence correction: Identify why an article is required, incorrect or unnecessary.

• Read in context: Articles are easier to understand when studied in complete sentences rather than as isolated rules.

• Review your mistakes: Maintain a list of recurring grammar errors and revisit it during revision.

CAT 2026 Preparation Resources

Use these CAT eBooks and sample papers to practise Quantitative Aptitude, DILR and VARC. Select resources according to the topics you need to strengthen.

HCF and LCM questions reward a clear understanding of number properties, radar graph questions require accurate data interpretation, and article-based grammar questions test attention to sentence structure. Revise the formulas and rules, practise questions under timed conditions, and review mistakes to improve your preparation for CAT 2026.

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