Problems & Puzzles: Puzzles

Puzzle 1151 Follow-up to Puzzle 1150 Emmanuel Vantieghem proposed the following follow-up to Puzzle 1150:    Q1 : a) The two numbers should be coprime            b) The two numbers should be primes    Q2 : The numbers should be pairwise coprime.

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During the week 18-25 Nov. 2023. contributions came from Giorgos Kalogeropoulos, J-M Rebert, M. Branicky, G. Gusev, Shyam S. Gupta, Oscar Volpatti, Emmanuel Vantieghem

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Giorgos wrote:

Q1

a) 8753 * 96421 = 843973013

b) 8641 * 97523 = 842696243

 

Q2

2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 = 100
 

2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23  = 223092870

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Rebert wrote.

Q1 : a) 843973013 = 8753 * 96421

Q1 : b) 841788323 = 863 * 975421

 

Q2 :

99 = 1 * 99

194 = 1 * 2 * 97

1365 = 1 * 3 * 5 * 91

2670 = 1 * 2 * 3 * 5 * 89

84315 = 1 * 3 * 5 * 7 * 11 * 73

164010 = 1 * 2 * 3 * 5 * 7 * 11 * 71

10975965 = 1 * 3 * 5 * 7 * 11 * 13 * 17 * 43

223092870 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23

 

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Michael wrote.

Q1a:  843973013 = 96421 * 8753
Q1b:  842696243 = 97523 * 8641
 

Q2:    223092870 = 23 * 19 * 17 * 13 * 11 * 7 * 5 * 3 * 2

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Gusev wrote.

Q1:

a) 96421 * 8753 = 843973013 (coprime)

b) 8641 * 97523 = 842696243 (both primes)

 

Q2:

8 * 9 * 11 * 13 * 17 * 19 * 23 = 76488984 (pairwise coprime) and 8 + 9 + 11 + 13 + 17 + 19 + 23 = 100

 

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Shyam wrote.

Q1: a) Two numbers which are coprime and collectively using each of the
digits 1,2,3,4,5,6,7,8 and 9 only once, such that the product of the two
numbers is the largest possible are 8753 and 96421, i.e.,
8753 * 96421 = 843973013.
The second largest product is obtained by two coprime numbers 9643 and
87521, i.e., 9643 * 87521 = 843965003.
b) Two prime numbers which collectively using each of the digits
1,2,3,4,5,6,7,8 and 9 only once, such that the product of the two prime
numbers is the largest possible are 8641 and 97523, i.e.,
8641 * 97523 = 842696243
The second largest product is obtained by two prime numbers 9623 and
87541, i.e., 9623 * 87541 = 842407043

Q2: The largest product which can be obtained from numbers whose sum is
100 and the numbers are pairwise coprime is 223092870.
2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 = 100 and 2*3*5*7*11*13*17*19*23 =
223092870.

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Oscar wrote:

Q1a:
843973013 = 8753 * 96421


Q1b:
842696243= 8641 * 97523


Q2:
223092870 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23.

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Emmanuel wrote:

Q1 a Numbers : 8753 and 96421. Product : 843973013

Q1 b Primes : 8641 and 97523. Product :  842696243
Q2 Numbers : 2, 3, 5, 7, 11, 13, 17, 19, 23 (the first 9 primes), Product 223092870.

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