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Number theory is often the part of the Indian Olympiad Qualifier in Mathematics (IOQM) where a well-prepared student can pick up marks quickly, because the same few ideas come back every year. The trouble is that school maths rarely teaches them. This post works through three number theory questions from the real IOQM 2024 paper, names the pattern behind each, and ends with a practice routine.
What does an IOQM number theory question look like?
The HBCSE Mathematical Olympiad brochure for 2026-27 describes IOQM as a three-hour exam with 30 questions, each with an integer answer from 00 to 99. The official IOQM 2024 answer key groups questions as 2-point, 3-point and 5-point problems, ten of each. The IOQM site says in its FAQ that there was no negative marking in IOQM 2026 and that calculus and statistics are excluded.
For topics, the Syllabus for Mathematical Olympiad on the IOQM site lists divisibility and the division algorithm, gcd and the Euclidean algorithm, the fundamental theorem of arithmetic, congruences, Fermat's little theorem and Euler's phi function, among others. Check the current year's pages, since formats can change.
Pattern 1: factorials and primes
IOQM 2024, Question 1 in the official paper: find the smallest positive integer that does not divide 1 × 2 × ... × 9.
Every number from 1 to 9 is a factor of 9!. So is 10, because 9! contains both 2 and 5. The next number, 11, is prime and larger than 9, so it cannot appear in the product. The answer is 11, which matches the official key.
The idea: a prime p divides n! only if p ≤ n. Composite numbers divide n! whenever their prime factors can be found inside it.
Pattern 2: last two digits as a remainder mod 100
Question 3 asks for the last two digits of 5²⁰²⁴. The last two digits of a number are its remainder when divided by 100. Start small: 5² = 25 and 25 × 5 = 125, which ends in 25 again. Once a power of 5 ends in 25, multiplying by 5 gives a number ending in 25 (since 25 × 5 = 125). So every power from 5² on ends in 25, and the answer is 25.
For a base like 7, look for a power that is 1 mod 100: 7⁴ = 2401. Since 2026 = 4 × 506 + 2, 7²⁰²⁶ ends in the same two digits as 7², which is 49.
Pattern 3: counting divisors by pairing
Question 29: let n = 2¹⁹ × 3¹². How many positive divisors of n² are less than n but do not divide n? Give the last two digits.
n² = 2³⁸ × 3²⁴ has (38 + 1)(24 + 1) = 975 divisors.
Pair each divisor d with n²/d. In every pair one number is below n and one is above, except d = n itself. So (975 − 1)/2 = 487 divisors of n² are less than n.
Every divisor of n is also a divisor of n². The number n has (19 + 1)(12 + 1) = 260 divisors, of which 259 are less than n.
The required count is 487 − 259 = 228. The last two digits are 28, which matches the official key.
Our post on counting divisors covers the (a + 1)(b + 1) formula in more detail.
Common mistakes and how to avoid them
Stopping at the first "obvious" answer. In Question 1, students often answer 10 without checking that 2 × 5 is inside 9!.
Forgetting the unpaired divisor. In pairing arguments, the square root of n² pairs with itself, so subtract it before halving.
Writing the answer in the wrong form. If a question asks for the last two digits, 228 must be entered as 28.
Guessing a cycle without checking it. Compute the powers until the remainder repeats, and write the cycle length down.
Why study solved questions before solving new ones?
In a series of experiments, Sweller and Cooper (1985) found that students who studied worked algebra examples processed them faster than students who solved the same problems, and then solved similar problems more quickly and with fewer errors. The benefit was specific to problems with the same structure. That is why naming the pattern matters: it lets you see that a new question has the same structure as one you have studied.
A practice routine for IOQM number theory
Download a past paper from the HBCSE past papers page and sort its questions into patterns: primes and factorials, remainders, divisors, gcd, equations in integers.
For each pattern, study one official solution, then solve two fresh questions of the same type without notes.
Once a week, do a mixed set of six number theory questions in 45 minutes with no topic labels. Rohrer, Dedrick and Stershic found that 126 seventh-graders who practised mixed problem types scored higher on tests one day and 30 days later than those who practised one type at a time.
Keep a log of the idea that cracked each problem, such as "pair the divisors" or "find a power that is 1 mod 100".
When a solution uses an idea you have not met, such as congruences, learn it properly before moving on. The Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, so you can look up the idea and come back to the problem.
Next step: find the remainder when 2²⁰²⁶ is divided by 7, using the fact that 2³ = 8 leaves remainder 1, then check your answer of 2.
