From the beginning of human history, prime numbers aroused human curiosity. What are they? Why are the questions related to them so hard? One of the most interesting things about prime numbers is their distribution among the natural numbers. On a small scale, the appearance of prime numbers seems random, but on a large scale there appears to be a pattern, which is still not fully understood.

Here are some interesting questions from prime numbers. Go ahead and play with these questions. Happy solving!

**Question 1: We want the smallest 5-digit snowball prime.**

## Solution

A snowball prime means that if one starts removing the digits of the number from right, the number would still be a prime.

So let’s start off with 2 (a prime number) and then to make it to 5 digit number first made a 2-digit prime number i.e,I just put another 2 , so it will become 22 . Now since 22 is not a prime number, then go for the immediate next prime number which is 23.

Then add one more 2 at the end of this number – then this number will become 232. Once again, as 232 is not a prime number move to the immediate next prime number which is available – it is 233.

By now you’ve realised the pattern,right?

To make it a 4 digit prime number just put another 3 at the end of this number, so it will become 2333.

Finally, to make it a 5 digit prime number put another 3 at the end, and here you go!

The smallest 5-digit snowball prime is 23333

## Question 2

**There are 2-digit prime numbers ‘ab’ such that the prime number is the (a*b)th prime number. For instance, 17 is the 7 ^{th} prime number and 73 is the 21^{st} prime number. This is why some folks call 73 as the best number in the world, but that’s a discussion for another day. There is no three digit number ‘abc’ that is (a*b*c)th prime. However, there is/are perhaps one(or more) 3-digit number(s) ‘abc’ that is the (a*b*c)th odd prime. Give all possible such 3-digit prime numbers and pat yourself on the back.**

## Solution

0 can never be a prime number so checking for all those primes with zeroes would not add to the result.

Each number from 100-146(excluding 139 whose product is 27 but is the 33 odd prime) are not possible as there are 24 single and double digit primes and the maximum product in these numbers is 24.

Primes between 147 and 200 can be considered and range from 34 th odd prime number to 45 the odd prime.

Another important point to note is that the numbers cannot have 2 1s as digits as it would lead to only a single digit product.It cannot have 2 2s as the maximum product is 36 and prime numbers from 100 -200 are numbered well beyond that range.

In other words any combination of digits that yield a single digit product and double digit product less than 25 are not considered.

Considering the factors above we can eliminate a lot of brute force checks.

Number of primes

Below 200- 46

Below 300 -61

Below 400 -77

Below 500 – 94

Below 600- 108

Below 700 -124

Below 800- 138

Below 900 -153

Below 1000 -167

First digit 1– product ranging between 31 to 46(considering few factors mentioned above).

Number product position

149 product 36 position 34

157 product 35 position 36

167 38th but product 42

179 63 is huge jump

197 and 199 also do not fall

So no prime of the desired requirement below 200

Between 200 to 300 we only need to look at even oddth primes as 2 would be one of the factors and must range between 47 and 61

48

50

52

54

56

58

60

48=2*24 283 is 60th prime

50 2*25 5*5 255 not prime

52 2*26 2*13*2 13 2 digit prime not possible

54 2*27 2*9*3 or 2*3*9

239 is 51st and 293 61 st so not possible

56 2*4*7 not prime

58 2*29 not possible

60 not possible

Therefore no luck between 200-300

61-74 multiples of 3

63

Possibility

373 73 rd

337 67th

66

3*11*2 not possible as 11 is 2 digit prime number

69 3*23 not possible as 23 is 2 digit prime

72 3*8*3 76th prime

Not found between 300-400

With the similar format ,hence forth,

400-500

80 84 88 92

80 4*4*5 not prime

84 4*7*3 or 4*3*7not primes

88 4*11*2 not possible

92 4*23 not possible

Not found in 400-500

500-600

95 100 105

95 5*19 not possible

100 5*5*4 not prime

105 5*5*7 101 the prime

So not found in this range

600-700

108 114 120

108 6*2*9 not prime

114 6*19 not possible

120 6*4*5 not prime

Not found

700-800

126 7*2*9 not prime

133 7*19 not possible

800-900

144 8*2*9 prime and 144th odd prime so 829 is one such prime

152 8*19 not possible

900-1000

153 9*17 not possible

162 9*2*9 157th odd prime

So only 829 is one such prime which is 144th odd prime which is 8*2*9.

Best wishes for CAT 2021!!

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