Let a number of three digits have for its middle digit the sum of the other two digits. Then it is a multiple of—

1. Place Value and Number Expansion (basic)

In the realm of Quantitative Aptitude, the most fundamental building block is understanding that a number is not just a collection of digits, but a sum of values based on position. Consider a three-digit number like 412. While the face values of the digits are 4, 1, and 2, their place values depend on their specific column. This concept is very similar to how we calculate the net power of lenses in physics by taking an algebraic sum of individual components (Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.158). In our decimal system, each position represents a power of 10. To represent any three-digit number algebraically, let the digits be 'a' (hundreds), 'b' (tens), and 'c' (units). We express this number as 100a + 10b + c. This is known as Number Expansion. For instance, if you have a Treasury Bill with a face value of ₹100 (Indian Economy, Nitin Singhania (ed 2nd 2021-22), Agriculture, p.259), the digit '1' carries the weight of a 'hundred' because of its position. If we move that '1' to the units place, its value changes drastically, even though the digit itself remains '1'. Understanding expansion allows us to manipulate numbers mathematically. If a problem tells you that the "tens digit is twice the units digit," you can translate that directly into an equation (b = 2c) and substitute it into your expansion (100a + 10(2c) + c). This transformation from a description of a number to an algebraic expression is the secret to solving complex digit-based puzzles in the UPSC CSAT.

Sources: Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.158; Indian Economy, Nitin Singhania (ed 2nd 2021-22), Agriculture, p.259

2. Classification of Numbers and Integers (basic)

To master Quantitative Aptitude, we must first understand the 'genealogy' of numbers. Just as the Indian Constitution allows for reasonable classification based on specific criteria like geography, time, or occupation Introduction to the Constitution of India, FUNDAMENTAL RIGHTS AND FUNDAMENTAL DUTIES, p.103, mathematics organizes numbers into distinct categories based on their inherent properties. At the top level, we deal with Real Numbers, which are divided into Rational Numbers (fractions and repeating decimals) and Irrational Numbers (non-repeating, non-terminating decimals like √2 or π).

Focusing on the building blocks of most aptitude problems, we find the Integers (Z). These are whole numbers that can be positive, negative, or zero {...-3, -2, -1, 0, 1, 2, 3...}. Within this set, we classify numbers into specific 'zones' of identity, much like the Indian Railways is classified into zones such as the Southern or Central zones for administrative efficiency Geography of India, Transport, Communications and Trade, p.15.

Category	Description	Examples
Natural Numbers (N)	Counting numbers starting from 1.	1, 2, 3, 4...
Whole Numbers (W)	Natural numbers plus zero.	0, 1, 2, 3...
Prime Numbers	Numbers > 1 with exactly two factors (1 and itself).	2, 3, 5, 7, 11...
Composite Numbers	Numbers with more than two factors.	4, 6, 8, 9, 10...

One of the most critical concepts for higher-level problems is the Algebraic Representation of these numbers. While we see the number 352 as a single entity, its mathematical structure is (100 × 3) + (10 × 5) + 2. In general terms, any three-digit number with digits a, b, and c is expressed as 100a + 10b + c. This expanded form is the key to proving divisibility rules and solving digit-based puzzles.

Sources: Introduction to the Constitution of India, FUNDAMENTAL RIGHTS AND FUNDAMENTAL DUTIES, p.103; Geography of India, Transport, Communications and Trade, p.15

3. Basic Divisibility Rules (2, 3, 5, 9) (intermediate)

In the UPSC CSAT, speed and accuracy are your best allies. Divisibility rules are not just shortcuts; they are foundational tools that allow you to analyze large numbers without performing tedious long division. To understand these rules from first principles, we must look at how numbers are constructed. Any three-digit number 'abc' can be expressed in its expanded algebraic form as 100a + 10b + c. This logic of using an algebraic sum to represent a whole is a common principle across disciplines, whether you are calculating the net power of combined lenses Science Class X, Light – Reflection and Refraction, p.158 or breaking down a complex number into its constituents.

The rules for 2 and 5 are the most intuitive because they depend entirely on the last digit. This is because 10, 100, and 1000 are all perfectly divisible by both 2 and 5. Therefore, in the expression 100a + 10b + c, the terms '100a' and '10b' will always be divisible by 2 and 5, regardless of what 'a' and 'b' are. The divisibility of the entire number depends solely on the units digit (c). If 'c' is even, the number is divisible by 2. If 'c' is 0 or 5, it is divisible by 5.

The rules for 3 and 9 require a slightly different algebraic perspective. We can rewrite 100a + 10b + c as (99 + 1)a + (9 + 1)b + c. Distributing this gives us (99a + 9b) + (a + b + c). Since 99 and 9 are multiples of both 3 and 9, the term (99a + 9b) is always divisible by them. Thus, the whole number is divisible by 3 or 9 only if the remaining part—the sum of its digits (a + b + c)—is also divisible by 3 or 9 respectively. This is why we check the total sum of digits to determine divisibility for these two specific divisors.

Divisor	Condition for Divisibility	Algebraic Logic
2	Last digit is even (0, 2, 4, 6, 8).	10n is always divisible by 2; only the units place matters.
5	Last digit is 0 or 5.	10n is always divisible by 5; only the units place matters.
3	Sum of digits is divisible by 3.	Powers of 10 leave a remainder of 1 when divided by 3.
9	Sum of digits is divisible by 9.	Powers of 10 leave a remainder of 1 when divided by 9.

Sources: Science Class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.158

4. Unit Digit and Cyclicity (intermediate)

In the realm of Quantitative Aptitude, the Unit Digit is the digit in the one's place of a number. For UPSC aspirants, mastering the unit digit is not about long multiplication, but about identifying patterns. When we raise a number to a high power, the last digit of the result follows a repeating sequence known as Cyclicity. Just as the economy undergoes fluctuations through Cyclical Unemployment—where trends repeat based on the business cycle peak and trough Vivek Singh, Indian Economy, Inclusive growth and issues, p.272—mathematical units follow a rhythmic repetition that allows us to predict the end of a massive calculation without performing it entirely. To find the unit digit of any expression like xⁿ, we only care about the unit digit of the base 'x'. These digits fall into three primary groups based on their cycle length:

• Cycle of 1: Digits {0, 1, 5, 6} always result in themselves. For example, any power of 5 ends in 5, and any power of 6 ends in 6.
• Cycle of 2: Digits {4, 9} repeat every two steps. 4ⁿ ends in 4 if n is odd, and 6 if n is even. Similarly, 9ⁿ ends in 9 if n is odd, and 1 if n is even.
• Cycle of 4: Digits {2, 3, 7, 8} have a four-step pattern. For instance, the powers of 2 end in 2, 4, 8, 6 before returning to 2.

To simplify any problem, use the Rule of 4: Divide the exponent by 4 and find the remainder (R). If the remainder is 1, 2, or 3, the unit digit is (unit digit of base)ᴿ. If the remainder is 0 (meaning the power is perfectly divisible by 4), we use the 4th power in the cycle. This systematic approach to breaking down complex data into manageable 'units' is a vital skill, much like how resource capacities are cataloged unit-by-unit in national energy planning Majid Hussain, Environment and Ecology, Distribution of World Natural Resources, p.25. For example, to find the unit digit of 7³³, we divide 33 by 4, getting a remainder of 1. Thus, the unit digit is 7¹ = 7.

Sources: Indian Economy, Inclusive growth and issues, p.272; Environment and Ecology, Distribution of World Natural Resources, p.25

5. Remainder Theorem and Applications (exam-level)

At its heart, the Remainder Theorem is an elegant shortcut. In basic arithmetic, we know that when we divide a dividend by a divisor, the 'leftover' part is the remainder. Mathematically, this is expressed as: Dividend = (Divisor × Quotient) + Remainder. In competitive exams, we extend this to algebra: if a polynomial p(x) is divided by a linear factor (x - a), the remainder is simply the value of the polynomial when x is replaced by a, i.e., p(a). This allows us to find remainders without performing long division, saving precious time during the exam. Applications of this theorem frequently appear in Number Theory, particularly when analyzing the structure of multi-digit numbers. For instance, any three-digit number 'abc' can be expanded algebraically as 100a + 10b + c. By manipulating this expression, we can derive divisibility rules. For example, if a number's digits follow a specific pattern—such as the middle digit being the sum of the outer two (b = a + c)—the algebraic form simplifies to 110a + 11c, or 11(10a + c). Because this expression is a clean product of 11, the remainder when divided by 11 is zero. This logic is the foundation for the 'alternating sum' rule used to check divisibility by 11. While the term 'remainder' is a mathematical staple, it is interesting to note its conceptual parallel in Indian Polity. Just as a mathematical remainder is the part that survives a division, the term 'remainder' is used in the Panchayati Raj system to describe the leftover duration of a dissolved body. If a Panchayat is dissolved early, the newly elected body does not start a fresh five-year term; it serves only for the remainder of the period for which the original body would have continued Indian Polity, Panchayati Raj, p.389. Understanding these properties is crucial because they allow us to handle large numbers and complex equations by breaking them into smaller, manageable parts. Below is a quick summary of how remainders behave under different operations:

Operation	Rule for Remainders
Addition	The remainder of a sum is the sum of the individual remainders.
Multiplication	The remainder of a product is the product of the individual remainders.
Powers	The remainder of (Aⁿ) divided by 'k' is the same as (Remainder of A/k)ⁿ.

Sources: Indian Polity, Panchayati Raj, p.389; Introduction to the Constitution of India, FUNDAMENTAL RIGHTS AND FUNDAMENTAL DUTIES, p.100

6. The Divisibility Rule for 11 (exam-level)

To master the Divisibility Rule for 11, we must move beyond simple division and look at the internal logic of how numbers are constructed. The rule states that a number is divisible by 11 if the difference between the sum of the digits in odd positions and the sum of the digits in even positions is either 0 or a multiple of 11. This isn't just a trick; it is rooted in how we represent numbers in base-10. For example, in a three-digit number abc, the value is 100a + 10b + c. We can logically decompose this into (99a + a) + (11b - b) + c. By grouping the parts clearly divisible by 11, we get 11(9a + b) + (a - b + c). Just as the total potential difference in a series circuit is the sum of individual differences Science, Class X (NCERT 2025 ed.), Electricity, p.183, the divisibility of the whole number depends entirely on whether that final sum (a - b + c) is divisible by 11.

A very common pattern seen in competitive exams involves three-digit numbers where the middle digit is exactly the sum of the outer two (b = a + c). Using our rule, the alternating sum would be a - (a + c) + c, which simplifies to 0. Since 0 is divisible by 11, any number following this pattern (like 132, 253, or 495) is guaranteed to be a multiple of 11. This algebraic certainty is a powerful tool for quick calculations under pressure.

Sources: Science, Class X (NCERT 2025 ed.), Electricity, p.183

7. Solving the Original PYQ (exam-level)

This question is a classic application of place value notation and divisibility rules, concepts you’ve just mastered. By representing the three-digit number algebraically as 100a + 10b + c, we can see how the internal structure of a number dictates its properties. The problem provides a specific constraint: the middle digit is the sum of the others (b = a + c). When you substitute this constraint into our algebraic model, the expression transforms into 100a + 10(a + c) + c, which simplifies to 110a + 11c. By factoring out the common 11, we arrive at 11(10a + c), mathematically proving that any number meeting this condition must be a multiple of 11.

As a coach, I want you to also see the shortcut logic: think back to the alternating sum rule for divisibility by 11. For a number 'abc' to be divisible by 11, the result of (a - b + c) must be 0 or a multiple of 11. Since the question states the middle digit is the sum of the outer two (b = a + c), then the operation (a + c) - b will always equal zero. This makes (A) 11 the logically inevitable answer. This approach saves precious seconds during the CSAT paper by bypassing long-form algebra.

UPSC often includes distractors like Option (B) 10 and Option (D) 50 to catch students who reflexively think of round numbers or specific endings; however, a number like 121 (where 1+1=2) satisfies the condition but is clearly not divisible by 10 or 50. Option (C) 18 is a complexity trap; while some numbers like 495 might fit certain patterns, a multiple of 18 must be even and divisible by 9, which is not a universal property for every number where the middle digit is the sum of the others.