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73 is the largest two-digit number that is prime and has prime numbers for both of its digits.
23
The only two digit numbers whose digits add up to 17 are 89 and 98. Since 98 is an even number, 89 is prime.
The smallest two-digit prime number whose digits themselves are prime is 23. Both 2 and 3 are prime numbers, and together they form the prime number 23.
23