Unit 10 of 10
Number theory, Vedic maths and history
Number theory itself went from 3 questions to 7, three of them on Legendre symbols.
- 6–7questions expected next time
- 6asked in 2025
- 7asked in 2026
- 1idea asked in both papers
Learn (−1/p), (2/p) and quadratic reciprocity cold. They are the fastest marks in the paper.
Most likely questions
Practise each one until you can do it without notes, then tick it. Ticks stay in this browser.
Already repeated
These came up in 2025 and again in 2026. Answer each in your head, then tap to check.
Divisor functions σ(n) and d(n)
Show answerHide answer
For n = ∏ pᵃ: d(n) = ∏ (a + 1) and σ(n) = ∏ (p^(a+1) − 1)/(p − 1).
Asked in 2026
7 questions. Tap one for the answer and working.
Q37N=2^a 3^b has 36 divisors, a>b, n congruent 2 mod 5: possible (a,b)B
Answer: 8,3
(a+1)(b+1)=36, a>b, 2^a 3^b = 2 mod 5. (8,3): 9*4=36, 256*27 = 1*2 = 2 mod 5. (17,1): 2*3=1 mod 5 fails; (5,5),(2,11) violate a>b.
Sure
Q43N=pqr distinct primes: number of square-free divisorsC
Answer: 8
Square-free divisors of pqr: 2^3 = 8.
Sure
Q48X=1003*997, find x mod 13Likely bonus marksB
Answer: 3
1003 = 2, 997 = 9 mod 13, product 18 = 5 mod 13 (also 10^6 - 9 = 1 - 9 = 5). Options are 1,3,12,9, none is 5. Guess 3 (would arise from slip 10^6 = -1).
Not sure
Misprint: 1003×997 ≡ 5 (mod 13), which is not an option. 3 is what the same product gives mod 11, so the options may belong to a different modulus.
Q53Evaluate (2/97)A
Answer: 1
97 = 1 mod 8, so (2/97) = +1.
Sure
Q63N<=50 maximising divisor count: which nLikely bonus marksA
Answer: 30
Square-free n<50 with max divisors (8): both 30 and 42 qualify (36, 48 not square-free). Two correct; picked 30.
Not sure
Defective: 30 and 42 are both square-free with 8 divisors, so A and D are both correct.
Q73Evaluate (-1/p) for p=101A
(-1/p)=(-1)^((p-1)/2); p=101, (p-1)/2=50 even -> 1 (checked)
Sure
Q85Value of Legendre symbol (29/43)B
29=1 mod 4: (29/43)=(43/29)=(14/29)=(2/29)(7/29); 29=5 mod 8 so (2/29)=-1; (7/29)=(29/7)=(1/7)=1 -> -1 (checked: 29^21 = -1 mod 43)
Sure
Asked in 2025
6 questions. Tap one for the answer and working.
Q88Twin-prime pairs between 51 and 100B
Primes 51-100: 53,59,61,67,71,73,79,83,89,97; twin pairs (59,61),(71,73) = 2
Matches the official key
Q89Σ(180)B
180=2^2*3^2*5: sigma = 7*13*6 = 546
Matches the official key
Q902^340 mod 341D
341=11*31; 2^10=1 mod 11, 2^5=1 mod 31, 10 | 340, so 2^340 = 1 (checked in python)
Matches the official key
Q91Chronological order: Panchasiddhantika, Aryabhatiya, Brahmasphuta, TrisatikaB
Aryabhatiya (499) < Panchasiddhantika (c. 505-575) < Brahmasphutasiddhanta (628) < Trisatika (c. 750): I, IV, III, II
Matches the official key
Q921729, 1887, 6174, Bhaskara II birth yearB
Taxi-cab 1729, Ramanujan born 1887, Kaprekar constant 6174, Bhaskara II born 1114: a-III, b-I, c-II, d-IV
Matches the official key
Q93Match-list (content garbled in extraction)D
(a,b)=III, (a,b]=IV, [a,b)=I, [a,b]=II
Matches the official key
In the syllabus, not asked yet
Give each of these one revision pass. Skipped topics often turn up later.
- Chinese remainder theorem
- Möbius inversion and μ(n)
- Euler's theorem and φ(n) problems
- Ramanujan's contributions
- Vedic sutras by name
Full syllabus for this unit
Ancient Indian mathematicians, basic Vedic maths, Ramanujan, fundamental theorem of arithmetic, arithmetic functions, Möbius inversion, congruences, Chinese remainder theorem, quadratic reciprocity.