Official Syllabus Item: 19, 31, 32, 44
Arithmetic & Number Theory
Prime, Composite Numbers & Divisibility Tests
Master prime & composite numbers, co-primes, twin primes, prime triplets, and divisibility tests (2 to 11, 72, 88, 99) for RIMC Stage-I descriptive exam.
Overview & Fundamental Concept
Every natural number greater than 1 is either prime or uniquely factorizable as a product of primes by the Fundamental Theorem of Arithmetic. In RIMC descriptive mathematics, candidates are tested on rigorous definitions of primes, composites, co-prime pairs, twin primes, prime triplets, square root bounding for primality testing, and algebraic divisibility criteria from 2 through 11 as well as compound divisors (72, 88, 99).
Key Theoretical Principles
- 1 is neither prime nor composite because it has only one positive factor.
- 2 is the smallest prime number and the only even prime number.
- Two numbers are co-prime if gcd(a, b) = 1; they do not need to be prime themselves.
- Twin primes differ by 2; there are exactly 8 twin prime pairs under 100.
- The set (3, 5, 7) is the unique prime triplet of the form (p, p+2, p+4).
- To test composite divisibility (e.g. 72, 88), factor into strictly co-prime divisors.
Illustrated Conceptual Diagrams & Visual Schematics
Prime Factorization Tree & Fundamental Divisibility Matrix
Fundamental Theorem of Arithmetic: Every composite number can be uniquely expressed as a product of prime powers. Divisibility shortcuts: 2 (even last digit), 3 (sum of digits div by 3), 4 (last 2 digits div by 4), 6 (div by both 2 and 3), 8 (last 3 digits div by 8), 9 (sum of digits div by 9), 11 (alternating digit difference is 0 or multiple of 11).
Topper Strategy & Descriptive Step Breakdown
Mastering RIMC Mathematics Standard
Number places from the rightmost units digit when testing divisibility by 11. Tabulate S_odd and S_even in distinct lines. For missing digit problems, state the bound constraint 0 <= x <= 9 to justify the unique solution.
Step 1: Given Information & Variable Declaration (1-2 Marks)
Clearly write down the data provided in the question. State any assumption (e.g., 'Let the cost price be ₹ x').
Step 2: Formula & Conceptual Foundation (1-2 Marks)
State the exact mathematical theorem or formula in standard algebraic notation before entering numeric values.
Step 3: Systematic Calculation & Reduction (2-4 Marks)
Carry out each step on a fresh line. Avoid combining three operations in one step so examiners can award partial credit.
Step 4: Final Statement with Proper Units (1 Mark)
End with a concluding statement highlighting the answer in a neat box or underline with explicit units (e.g., '₹ 1,500', '45 km/h').
Key Formulas & Conversion Cheat Sheet
Divisibility by 3 & 9
$$3 \mid N \iff 3 \mid \sum d_i \quad \text{and} \quad 9 \mid N \iff 9 \mid \sum d_i$$
Divisibility by 4 & 8
$$4 \mid N \iff 4 \mid (10t + u); \quad 8 \mid N \iff 8 \mid (100h + 10t + u)$$
Divisibility by 11
$$11 \mid N \iff |S_{\text{odd}} - S_{\text{even}}| = 0 \text{ or } 11k \quad (k \in \mathbb{N})$$
Co-Prime Divisor Theorem
$$\text{If } \gcd(a, b) = 1, \quad [a \mid N \text{ and } b \mid N] \implies (a \times b) \mid N$$
Primality Criterion
$$k = \lceil\sqrt{N}\rceil; \quad (\forall p \le k, p \nmid N) \implies N \in P$$
Solved Model Examples (Descriptive Format with Step Mark Distribution)
Solved Model 1
Model Solution
Find the single-digit value of * such that the 7-digit number 5432*71 is completely divisible by 9.
Step-by-Step Model Solution:
Step 1: Rule for divisibility by 9: Sum of digits must be a multiple of 9.\nStep 2: Sum of digits = 5 + 4 + 3 + 2 + * + 7 + 1 = 22 + *.\nStep 3: Since * is a single digit, 0 <= * <= 9, so 22 <= 22 + * <= 31. The only multiple of 9 in this range is 27.\nStep 4: 22 + * = 27 => * = 27 - 22 = 5.\nFinal Answer: * = 5 (Complete number: 54,32,571).
Solved Model 2
Model Solution
Replace * with the smallest non-zero digit so that the number 86*725 is divisible by 11.
Step-by-Step Model Solution:
Step 1: Rule for 11: |S_odd - S_even| must be 0 or a multiple of 11 (places counted from right).\nStep 2: S_odd = 5 + 7 + 6 = 18; S_even = 2 + * + 8 = 10 + *.\nStep 3: Difference = 18 - (10 + *) = 8 - *. For 8 - * = 0 => * = 8.\nFinal Answer: * = 8 (Complete number: 868725).
Solved Model 3
Model Solution
A 6-digit number 4x572y is completely divisible by 72. Find the values of digits x and y, and evaluate (3x + 2y).
Step-by-Step Model Solution:
Step 1: 72 = 8 × 9 where gcd(8, 9) = 1.\nStep 2: For divisibility by 8, last three digits 72y must be divisible by 8 => 720 + y => y = 0 or 8.\nStep 3: For divisibility by 9, sum of digits 4+x+5+7+2+y = 18+x+y must be divisible by 9 => (x+y) in {0, 9, 18}. For non-zero pair y = 8 => x + 8 = 9 => x = 1.\nStep 4: 3x + 2y = 3(1) + 2(8) = 19.\nFinal Answer: x = 1, y = 8, and 3x + 2y = 19.
RIMC Practice & Drill Exercises
Attempt these questions on paper following proper step-wise conventions, then click Reveal Solution to verify your answers.
Question 1
5 Marks
Using the descriptive divisibility test, check whether the number 73,81,038 is divisible by 11.
Model Descriptive Solution:
S_odd = 8 + 0 + 8 + 7 = 23. S_even = 3 + 1 + 3 = 7. Difference = 23 - 7 = 16. Since 16 is not divisible by 11, 73,81,038 is not divisible by 11.
Question 2
5 Marks
Find all possible single-digit values of x such that 34x51 is divisible by 3.
Model Descriptive Solution:
Sum of digits = 3 + 4 + x + 5 + 1 = 13 + x. Since 0 <= x <= 9, 13 <= 13 + x <= 22. Multiples of 3 are 15, 18, 21. Thus x = 2, 5, or 8.
Question 3
5 Marks
Find the least natural number that must be subtracted from 1385 so that the remainder is divisible by 16.
Model Descriptive Solution:
1385 ÷ 16 = 86 with remainder 9. The least number to be subtracted is the remainder, which is 9.
Question 4
5 Marks
Find the least natural number that must be added to 45,156 to make it exactly divisible by 9.
Model Descriptive Solution:
Sum of digits = 4 + 5 + 1 + 5 + 6 = 21. Next multiple of 9 is 27. Number to be added = 27 - 21 = 6.
Question 5
5 Marks
Write all prime numbers between 90 and 100. Provide complete reasons why 91 is not prime.
Model Descriptive Solution:
The only prime between 90 and 100 is 97. 91 is composite because 91 = 7 × 13, so its factors are {1, 7, 13, 91} (more than 2 factors).
Question 6
10 Marks
If the 8-digit number 342x18y6 is divisible by 88, find distinct single digits x and y (x != y).
Model Descriptive Solution:
88 = 8 × 11. Last three digits 8y6 divisible by 8 => y in {1, 5, 9}. Divisibility by 11 gives S_odd - S_even = (18+x) - (6+y) = 12 + x - y = 11 => x = y - 1. For y = 5, x = 4 (or y = 1, x = 0; y = 9, x = 8).
Question 7
10 Marks
Check whether 571 is a prime number, listing all prime divisors tested with square root bounds.
Model Descriptive Solution:
24^2 = 576 > 571. Test primes <= 24: 2, 3, 5, 7, 11, 13, 17, 19, 23. None divides 571. Hence 571 is a prime number.
Question 8
5 Marks
Prove algebraically that any two consecutive natural numbers n and n+1 are always co-prime.
Model Descriptive Solution:
Let d divide both n and n+1. Then d divides (n+1) - n = 1. Since d is a positive integer, d = 1. Thus gcd(n, n+1) = 1. Hence proved.
Question 9
10 Marks
A 5-digit number 517x3 is divisible by 3 and NOT divisible by 9. Find all possible single-digit values of x.
Model Descriptive Solution:
Sum of digits = 16 + x. For divisibility by 3, 16+x in {18, 21, 24} => x in {2, 5, 8}. For 16+x NOT divisible by 9, reject x = 2 (sum 18). Accepted values: x = 5 and x = 8.
Question 10
10 Marks
Show that the sum of any three consecutive even natural numbers is always divisible by 6.
Model Descriptive Solution:
Let numbers be 2k-2, 2k, 2k+2. Sum = (2k-2) + 2k + (2k+2) = 6k = 6 × k. Since k is an integer, the sum is a multiple of 6. Hence proved.
High-Yield FAQs & Revision Notes
Why is 1 neither prime nor composite?
By definition, a prime number has exactly two distinct positive divisors (1 and itself), while a composite number has more than two. 1 has only one positive divisor (itself), so it is neither.
What is the only even prime number, and why are there no other even primes?
2 is the only even prime number. Any other even number is divisible by 1, 2, and itself, giving it at least 3 divisors and making it composite.
What is the difference between co-prime numbers and twin primes?
Co-primes have gcd = 1 and need not be prime individually (e.g. 8 and 9). Twin primes are two primes differing by 2 (e.g. 11 and 13).
What is the unique prime triplet, and why can no other prime triplet exist?
The unique prime triplet (p, p+2, p+4) is (3, 5, 7). Among any p, p+2, p+4, one is always a multiple of 3. The only prime multiple of 3 is 3 itself.
How do we test if a number is divisible by composite divisors like 72, 88, or 99?
Resolve the composite divisor into two co-prime factors: for 72 use 8 and 9; for 88 use 8 and 11; for 99 use 9 and 11.
What is the method to determine whether a large number N is prime?
Find k such that k^2 >= N. Test divisibility only by primes <= k. If none divides N, then N is prime.
What are all the twin prime pairs between 1 and 100?
There are 8 pairs: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), and (71, 73).
What is Goldbach's conjecture and how is it tested in school mathematics?
Every even integer greater than 2 can be written as the sum of two primes. In RIMC, students express numbers like 84 as 41 + 43 or 44 as 3 + 41.
Why is the divisibility test for 11 based on alternating digit sums?
Because 10^k = (-1)^k mod 11, powers of 10 alternate between +1 and -1 modulo 11.
How does this module correspond to the official RIMC syllabus?
It directly covers Serial No. 19, 31, 32, and 44 of Chapter 8 in the official RIMC brochure.
How does this topic relate to the official RIMC syllabus?
This module directly covers Serial No. 19, 31, 32, 44 of the broader syllabus published in Chapter 8 of the official RIMC admission brochure, ensuring 100% syllabus alignment.