Determine the number of three-digit numbers with the following property: The number formed by the first two digits is prime and the number formed by the last two digits is prime.
Problem
Source:
Tags: number theory, Digits
BackToSchool
19.05.2024 23:42
We have $21$ two-digits primes $$11(4), 13(2), 17(3), 19(1), 23(2), 29(1), 31(4), 37(3), 41(4), 43(2), 47(3), 53(2), 59(1), 61(4), 67(3), 71(4), 73(2), 79(1), 83(2), 89(1), 97(3)$$The number in brackets are leading digit of two-digits primes and adds up to $\boxed{51}$.
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miyukina
19.05.2024 23:55
List of two-digit primes, without evens and 5
= { 11, 13, 17, 19, 31, 37, 71, 73, 79, 97 }
So four starts with 1, two with 3, three with 7 and one with 9
The left out two-digit primes
= { 23, 29, 41, 43, 47, 53, 59, 61, 67, 83, 89 }
In total there are five that ends with 1, six with 3, five with 7 and five with 9
Answer
= 5 × 4 + 6 × 2 + 5 × 3 + 5 × 1
= 20 + 12 + 15 + 5
= 52
AndrewTom
20.05.2024 10:42
I miscounted as usual. In much the same way as miyukina, I counted the ones with middle digits $1, 3, 7, 9$, getting $20 + 12 + 12 + 5 = 49$ but rushed and missed four of them. I realised I was wrong when I saw that miyukina had a different answer. Well done, miyukina.