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NCERT Solutions Ganita Prakash Chapter 5 & 115Section 5.2 Prime Numbers — Figure it Out

Book page 114 Updated on2026-09-05

Q1.
We see that 2 is a prime and also an even number. Is there any other even prime?
Answer

No. 2 is the only even prime number.

Take any even number bigger than 2, say 34.
34 = 2 × 17 → its factors are 1, 2, 17, 34 — that is four factors
Why it must be so: every even number has 2 as a factor. If the number is bigger than 2, then it has at least three factors — 1, 2 and the number itself. A prime is allowed only two factors, so no even number above 2 can be prime. The number 2 escapes because for 2 the “extra” factor 2 is the number itself.
Fun name: mathematicians call 2 “the oddest prime” — because it is the only even one!
Q2.
Look at the list of primes till 100. What is the smallest difference between two successive primes? What is the largest difference?
Answer

Write the primes and the gaps between them:

2 1 3 2 5 2 7 4 11 2 13 4 17 2 19 4 23 6 29 2 31 6 37 4 41 2 43 4 47 6 53
53 6 59 2 61 6 67 4 71 2 73 6 79 4 83 6 89 8 97

Smallest difference = 1, between 2 and 3.

Largest difference = 8, between 89 and 97.

3 − 2 = 1     97 − 89 = 8
Why 1 happens only once: two numbers that differ by 1 are one odd and one even. The only even prime is 2, so the only such pair can be 2 and 3. After that, all primes are odd, so every gap is an even number — 2, 4, 6 or 8.
Check it yourself: the numbers 90 to 96 in between 89 and 97 are all composite — that is the long stretch that creates the biggest gap.
Q3.
Are there an equal number of primes occurring in every row in the table on the previous page? Which decades have the least number of primes? Which have the most number of primes?
Answer

No, the primes are not spread out equally. Counting them row by row:

DecadePrimes in itHow many
1 – 102, 3, 5, 74
11 – 2011, 13, 17, 194
21 – 3023, 292
31 – 4031, 372
41 – 5041, 43, 473
51 – 6053, 592
61 – 7061, 672
71 – 8071, 73, 793
81 – 9083, 892
91 – 100971
4 + 4 + 2 + 2 + 3 + 2 + 2 + 3 + 2 + 1 = 25 primes ✔

Least: the decade 91 – 100, with only one prime (97).

Most: the decades 1 – 10 and 11 – 20, with four primes each.

Why primes thin out: as numbers grow bigger there are more and more smaller primes that could divide them, so it gets harder for a number to escape being composite. Primes never stop, but they do spread further apart.
Q4.
Which of the following numbers are prime: 23, 51, 37, 26?
Answer

For each number, look for a factor other than 1 and itself.

NumberCheckPrime or composite?
23not even; 2 + 3 = 5 (not a multiple of 3); does not end in 0 or 5; 4 × 4 = 16 < 23 < 25Prime ✔
515 + 1 = 6, a multiple of 3 → 51 = 3 × 17Composite
37not even; 3 + 7 = 10 (not a multiple of 3); does not end in 0 or 5; not a multiple of 7 eitherPrime ✔
26even → 26 = 2 × 13Composite

Answer: 23 and 37 are prime. 51 and 26 are composite.

Careful: 51 looks like a prime at first glance, but the digit-sum test catches it at once: 5 + 1 = 6, so 51 is in the 3 table.
Q5.
Write three pairs of prime numbers less than 20 whose sum is a multiple of 5.
Answer

The primes below 20 are 2, 3, 5, 7, 11, 13, 17, 19. Pick pairs whose sum ends in 0 or 5.

2 + 3 = 5 = 5 × 1 ✔
3 + 7 = 10 = 5 × 2 ✔
2 + 13 = 15 = 5 × 3 ✔

Three such pairs: (2, 3), (3, 7) and (2, 13).

There are more — try these too:

3 + 17 = 20 ✔    7 + 13 = 20 ✔    11 + 19 = 30 ✔    13 + 17 = 30 ✔
Try This: notice that every pair above except (2, 3) and (2, 13) is a pair of two odd primes, and their sum is always even. So an odd sum like 5 or 15 can only come from a pair that includes the even prime 2.
Q6.
The numbers 13 and 31 are prime numbers. Both these numbers have same digits 1 and 3. Find such pairs of prime numbers up to 100.
Answer

Take each two-digit prime, reverse its digits, and check whether the reversed number is also prime.

PrimeReversedIs the reverse prime?
1331Yes ✔
1771Yes ✔
3773Yes ✔
7997Yes ✔
1991 = 7 × 13No
2332 (even)No
2992 (even)No
5995 = 5 × 19No
8338 (even)No

Answer: the pairs are (13, 31), (17, 71), (37, 73) and (79, 97).

Did you know? Such primes are called emirps — “prime” written backwards! The number 11 reads the same both ways, so it is a palindromic prime rather than an emirp.
Q7.
Find seven consecutive composite numbers between 1 and 100.
Answer

Look for the biggest gap between two primes below 100. It comes between 89 and 97, and all the seven numbers in between are composite.

Number90919293949596
A factor2 × 457 × 132 × 463 × 312 × 475 × 192 × 48

Answer: 90, 91, 92, 93, 94, 95, 96.

Why 91 is the sneaky one: 90, 92, 94, 96 are even and 93, 95 are obviously in the 3 and 5 tables. Only 91 hides its factors — but 91 = 7 × 13, so it is composite too. That is what makes this a run of seven composites in a row, the longest one below 100.
Q8.
Twin primes are pairs of primes having a difference of 2. For example, 3 and 5 are twin primes. So are 17 and 19. Find the other twin primes between 1 and 100.
Answer

Run down the list of primes and note every pair that differs by 2.

(3, 5)   (5, 7)   (11, 13)   (17, 19)   (29, 31)   (41, 43)   (59, 61)   (71, 73)

The book already gives (3, 5) and (17, 19), so the other twin primes are

(5, 7), (11, 13), (29, 31), (41, 43), (59, 61) and (71, 73)

There are 8 twin-prime pairs below 100 in all.

Did you know? Nobody knows whether twin primes go on for ever. This is one of the most famous unsolved problems in mathematics, called the Twin Prime Conjecture.
Q9.
Identify whether each statement is true or false. Explain. a. There is no prime number whose units digit is 4. b. A product of primes can also be prime. c. Prime numbers do not have any factors. d. All even numbers are composite numbers. e. 2 is a prime and so is the next number, 3. For every other prime, the next number is composite.
Answer
StatementTrue / FalseReason
a. There is no prime number whose units digit is 4.TrueA number ending in 4 is even, so 2 is a factor of it. It is also bigger than 2, so it has at least three factors — 1, 2 and itself. Example: 14 = 2 × 7, 24 = 2 × 12.
b. A product of primes can also be prime.FalseMultiply two or more primes and the answer picks up those primes as extra factors. 3 × 5 = 15, whose factors are 1, 3, 5, 15 — four factors, so 15 is composite.
c. Prime numbers do not have any factors.FalseEvery prime has exactly two factors — 1 and itself. Factors of 13 are 1 and 13.
d. All even numbers are composite numbers.False2 is even but prime. Every other even number is indeed composite.
e. For every prime other than 2, the next number is composite.TrueEvery prime except 2 is odd, so the number just after it is even and bigger than 2 — hence composite. 3 → 4, 5 → 6, 7 → 8, 11 → 12, 97 → 98.
The common thread: statements (a), (d) and (e) all rest on the single fact that 2 is the only even prime. Once you hold on to that, all three become easy.
Q10.
Which of the following numbers is the product of exactly three distinct prime numbers: 45, 60, 91, 105, 330?
Answer

Find the prime factorisation of each and count the different primes in it.

NumberPrime factorisationHow many primes are multipliedExactly three distinct primes?
453 × 3 × 53 primes, but only 2 different onesNo
602 × 2 × 3 × 54 primes, 3 differentNo
917 × 132 primesNo
1053 × 5 × 73 primes, all differentYes ✔
3302 × 3 × 5 × 114 primes, all differentNo
3 × 5 × 7 = 15 × 7 = 105 ✔

Answer: 105.

Read the question carefully: 45 is a product of three primes but two of them are the same, and 330 has three different primes plus one more. Only 105 is a product of exactly three distinct primes.
Q11.
How many three-digit prime numbers can you make using each of 2, 4 and 5 once?
Answer

Answer: none — zero such primes.

The digits 2, 4 and 5 can be arranged in six ways:

Number245254425452524542
Last digit545242
Divisible by525222
245 = 5 × 49    254 = 2 × 127    425 = 5 × 85
452 = 2 × 226    524 = 2 × 262    542 = 2 × 271
Why not even one: whichever way you arrange them, the number must end in 2, 4 or 5. Ending in 2 or 4 makes it even (so 2 is a factor); ending in 5 makes it a multiple of 5. In every case the number has a factor other than 1 and itself, so it is composite.
Q12.
Observe that 3 is a prime number, and 2 × 3 + 1 = 7 is also a prime. Are there other primes for which doubling and adding 1 gives another prime? Find at least five such examples.
Answer

Yes, there are many. Double the prime and add 1, then check the answer.

Prime p2 × p + 1Is it prime?
22 × 2 + 1 = 5Yes ✔
32 × 3 + 1 = 7Yes ✔
52 × 5 + 1 = 11Yes ✔
112 × 11 + 1 = 23Yes ✔
232 × 23 + 1 = 47Yes ✔
292 × 29 + 1 = 59Yes ✔
412 × 41 + 1 = 83Yes ✔
72 × 7 + 1 = 15 = 3 × 5No
132 × 13 + 1 = 27 = 3 × 9No
172 × 17 + 1 = 35 = 5 × 7No

Five examples: 2 → 5, 5 → 11, 11 → 23, 23 → 47 and 29 → 59. (41 → 83 is a sixth.)

Did you know? Primes like 2, 3, 5, 11, 23, 29 and 41 — where doubling and adding 1 gives another prime — are called Sophie Germain primes, after the French mathematician Sophie Germain, who used them in her work on Fermat's Last Theorem.
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