Problems & Puzzles: Puzzles

 Puzzle 656 Primes using certain quantity of digits I will use one more time a puzzle (1002) from the interesting site of my friend Claudio Meller as a starting point to create another similar puzzle but related to prime numbers. BTW, congratulations to Claudio because his site recently arrived to 1000 entries! This puzzle goes like this: You are able to form distinct prime numbers using K times exactly the whole set of digits 1 to 9, such that the sum of all the primes formed is a minimal quantity. Example: Let's start with the simplest case: K=1. Then one solution is: {61, 283, 47, 59} Sum but not minimal = 450. If the set of allowed digits is 0 to 9 -zero leading in the primes is not allowed- then one solution is: {251, 409, 67, 83} Sum but not minimal = 810. Q1. Find the minimal solutions if you are able to use K times all the of the set of digits 1 to 9, for K=1 to 10 Q2. Redo Q1 using the set of digits 0 to 9.

Contribution came from Giovanni Resta.

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Resta wrote:

Q1 (zeroless)

1 : 207 = 2+3+5+47+61+89
2 : 477 = 2+5+29+41+47+53+61+67+83+89
3 : 1107 = 2+5+7+41+43+47+53+59+61+67+83+89+269+281
4 : 2034 = 5+29+41+43+47+53+59+61+67+83+89+157+269+281+283+467
5 : 3015 = 2+5+23+29+41+43+47+53+59+61+67+83+89+157+181+269+283+
467+487+569
6 : 4014 = 2+5+23+29+41+43+47+53+59+61+67+83+89+149+157+181+263+269+
283+467+487+569+587
7 : 5067 = 2+5+29+41+43+47+53+59+61+67+79+83+89+157+163+
181+257+263+269+281+283+449+463+487+569+587
8 : 6120 = 2+5+23+29+41+43+47+53+59+61+67+83+89+149+163+167+
181+257+263+269+281+283+389+449+467+487+557+569+587
9 : 7380 = 2+5+7+19+23+29+41+43+47+53+59+61+67+83+89+149+167+
181+251+257+263+269+281+283+389+449+463+467+487+557+
569+587+683
10 : 8622 = 2+5+29+41+43+47+53+59+61+67+79+83+89+149+157+163+167+
181+229+251+257+263+269+281+283+389+449+457+463+
467+487+563+569+587+883
11: 9909 = 2+5+29+41+43+47+53+59+61+67+79+83+89+149+157+163+167+181+
229+241+251+257+263+269+281+283+359+389+449+457+463+
467+487+563+569+587+683+887

Q2 (with zero)

1 :  567 = 2+5+7+61+83+409
2 : 1287 = 2+5+23+47+61+67+83+89+401+509
3 : 2187 = 2+5+23+47+53+59+61+67+83+89+281+401+409+607
4 : 3186 = 2+5+47+53+59+61+67+83+89+107+263+269+281+401+409+487+503
5 : 4266 = 2+5+41+43+47+53+59+61+67+83+89+107+263+269+281+283+401+
409+509+587+607
6 : 5535 = 2+5+43+47+53+59+61+67+83+89+103+109+251+263+269+281+283+
401+409+467+487+509+587+607
7 : 6795 = 2+5+29+41+43+47+53+59+61+67+83+89+107+109+251+263+269+281+
283+401+409+467+487+503+509+587+607+683
8 : 8325 = 2+5+7+29+41+43+47+53+59+61+67+83+89+107+109+251+257+263+269+
281+283+401+409+461+463+487+503+509+587+607+683+809
9 : 9684 = 2+5+23+29+41+43+47+53+59+61+67+83+89+103+107+109+251+257+
263+269+281+283+401+409+461+463+467+487+503+509+569+587+
607+809+887
10: 11223 = 2+5+29+41+43+47+53+59+61+67+83+89+101+103+107+109+227+
251+257+263+269+281+283+401+409+449+463+467+487+503+
509+563+569+587+607+683+809+887
11: 12870 = 2+5+29+43+47+53+59+61+67+83+89+103+107+109+181+229+241+
251+257+263+269+281+283+307+401+409+449+457+463+467+487+
503+509+563+569+587+601+607+683+809+887

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