{"id":14061,"date":"2021-07-27T16:45:47","date_gmt":"2021-07-27T11:15:47","guid":{"rendered":"https:\/\/online.2iim.com\/cat-exam\/blogs\/?p=14061"},"modified":"2021-07-27T16:45:48","modified_gmt":"2021-07-27T11:15:48","slug":"interesting-questions-from-number-system-cat-2021","status":"publish","type":"post","link":"https:\/\/online.2iim.com\/cat-exam\/blogs\/cat-quantitative-aptitude\/interesting-questions-from-number-system-cat-2021\/","title":{"rendered":"Interesting questions from Number System | CAT 2021"},"content":{"rendered":"\n
\"\"<\/figure>\n\n\n\n\n\n

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. <\/p>\n\n\n\n

Here are some interesting questions from prime numbers. Go ahead and play with these questions. Happy solving!<\/p>\n\n\n\n

Question 1: We want the smallest 5-digit snowball prime.<\/strong><\/h2>\n\n\n\n

Solution<\/h2>\n\n\n\n

A snowball prime means that if one starts removing the digits of the number from right, the number would still be a prime.<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

By now you’ve realised the pattern,right?<\/p>\n\n\n\n

To make it a 4 digit prime number just put another 3 at the end of this number, so it will become 2333.<\/p>\n\n\n\n

Finally, to make it a 5 digit prime number put another 3 at the end, and here you go!<\/p>\n\n\n\n

The smallest 5-digit snowball prime is 23333<\/p>\n\n\n\n

Question 2<\/h2>\n\n\n\n

There are 2-digit prime numbers \u2018ab\u2019 such that the prime number is the (a*b)th prime number. For instance, 17 is the 7th<\/sup>\u00a0prime number and 73 is the 21st<\/sup>\u00a0prime number. This is why some folks call 73 as the best number in the world, but that\u2019s a discussion for another day. There is no three digit number \u2018abc\u2019 that is (a*b*c)th prime. However, there is\/are perhaps one(or more)\u00a0 3-digit number(s) \u2018abc\u2019 that is the (a*b*c)th odd prime. Give all possible such 3-digit prime numbers and pat yourself on the back.<\/strong><\/p>\n\n\n\n

Solution<\/h2>\n\n\n\n

0 can never be a prime number so checking for all those primes with zeroes would not add to the result.<\/p>\n\n\n\n

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\u00a0 digit primes and the maximum product in these numbers is 24.<\/p>\n\n\n\n

Primes between 147 and 200 can be considered and range from 34 th odd\u00a0 prime number to 45 the odd prime.<\/p>\n\n\n\n

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\u00a0 numbers from 100 -200 are numbered well beyond that range.<\/p>\n\n\n\n

In other words any combination of digits that yield a single digit product and double digit product less than 25 are not considered.<\/p>\n\n\n\n

Considering the factors above we can eliminate a lot\u00a0 of brute force checks.<\/p>\n\n\n\n

Number of primes<\/p>\n\n\n\n

Below 200- 46<\/p>\n\n\n\n

Below 300 -61<\/p>\n\n\n\n

Below 400 -77<\/p>\n\n\n\n

Below 500\u00a0– 94<\/p>\n\n\n\n

Below 600- 108<\/p>\n\n\n\n

Below 700 -124<\/p>\n\n\n\n

Below 800- 138<\/p>\n\n\n\n

Below 900 -153<\/p>\n\n\n\n

Below 1000 -167<\/p>\n\n\n\n

First digit 1\u2013 product ranging between 31 to 46(considering few factors mentioned above).<\/p>\n\n\n\n

Number product position<\/p>\n\n\n\n

149 product 36 position 34<\/p>\n\n\n\n

157 product 35 position 36<\/p>\n\n\n\n

167 38th but product 42<\/p>\n\n\n\n

179 63 is huge jump<\/p>\n\n\n\n

197 and 199 also do not fall<\/p>\n\n\n\n

So no prime of the desired requirement below 200<\/p>\n\n\n\n

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<\/p>\n\n\n\n

48<\/p>\n\n\n\n

50<\/p>\n\n\n\n

52<\/p>\n\n\n\n

54<\/p>\n\n\n\n

56<\/p>\n\n\n\n

58<\/p>\n\n\n\n

60<\/p>\n\n\n\n

48=2*24 283 is 60th prime<\/p>\n\n\n\n

50 2*25 5*5 255 not prime<\/p>\n\n\n\n

52 2*26 2*13*2 13 2 digit prime not possible<\/p>\n\n\n\n

54 2*27 2*9*3 or 2*3*9<\/p>\n\n\n\n

239 is 51st and 293 61 st so not possible<\/p>\n\n\n\n

56 2*4*7  not prime<\/p>\n\n\n\n

58 2*29 not possible<\/p>\n\n\n\n

60 not possible<\/p>\n\n\n\n

Therefore no luck\u00a0 between 200-300<\/p>\n\n\n\n

61-74 multiples of 3<\/p>\n\n\n\n

63<\/p>\n\n\n\n

Possibility<\/p>\n\n\n\n

373\u00a0 73 rd<\/p>\n\n\n\n

337 67th<\/p>\n\n\n\n

66<\/p>\n\n\n\n

3*11*2 not possible as 11 is 2 digit prime number<\/p>\n\n\n\n

69 3*23 not possible as 23 is 2 digit prime<\/p>\n\n\n\n

72 3*8*3 76th prime<\/p>\n\n\n\n

Not found between 300-400<\/p>\n\n\n\n

With the similar format ,hence forth,<\/p>\n\n\n\n

400-500<\/p>\n\n\n\n

80 84 88 92<\/p>\n\n\n\n

80 4*4*5 not prime<\/p>\n\n\n\n

84 4*7*3 or 4*3*7not primes<\/p>\n\n\n\n

88 4*11*2 not possible<\/p>\n\n\n\n

92 4*23 not possible<\/p>\n\n\n\n

Not found in 400-500<\/p>\n\n\n\n

500-600<\/p>\n\n\n\n

95 100 105<\/p>\n\n\n\n

95 5*19 not possible<\/p>\n\n\n\n

100 5*5*4 not prime<\/p>\n\n\n\n

105 5*5*7 101 the prime<\/p>\n\n\n\n

So not found in this range<\/p>\n\n\n\n

600-700<\/p>\n\n\n\n

108 114 120<\/p>\n\n\n\n

108 6*2*9 not prime<\/p>\n\n\n\n

114 6*19 not possible<\/p>\n\n\n\n

120 6*4*5 not prime<\/p>\n\n\n\n

Not found<\/p>\n\n\n\n

700-800<\/p>\n\n\n\n

126 7*2*9 not prime<\/p>\n\n\n\n

133 7*19 not possible<\/p>\n\n\n\n

800-900<\/p>\n\n\n\n

144 8*2*9 prime and 144th odd prime so 829 is one such prime<\/p>\n\n\n\n

152 8*19 not possible<\/p>\n\n\n\n

900-1000<\/p>\n\n\n\n

153 9*17 not possible<\/p>\n\n\n\n

162 9*2*9 157th odd prime<\/p>\n\n\n\n

So only 829 is one such prime which is 144th odd prime which is 8*2*9.<\/p>\n\n\n\n

Best wishes for CAT 2021!!<\/p>\n\n\n\n\n","protected":false},"excerpt":{"rendered":"

Prime numbers have occupied human attention since ancient times and were even associated with the supernatural. Even today, in modern times, there are people trying to provide prime numbers with mystical properties. Here are two interesting questions based on Primes. Go ahead, play with these questions!<\/p>\n","protected":false},"author":1,"featured_media":14062,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[491,2],"tags":[235,173],"yoast_head":"\nInteresting questions from Number System | CAT 2021 - 2IIM CAT Preparation Blog<\/title>\n<meta name=\"description\" content=\"Prime numbers have occupied human attention since ancient times and were even associated with the supernatural. Even today, in modern times, there are people trying to provide prime numbers with mystical properties. Here are two interesting questions based on Primes. 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