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Removed some examples after helpful comment.
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Carl-Fredrik Nyberg Brodda
Carl-Fredrik Nyberg Brodda
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The article is [Blom, G. and Fröberg, C-E., Om Myntväxling, Nordisk Matematisk Tidskrift, 1962, Vol. 10, No. 1/2 (1962), pp. 55-69] for anyone who wishes to sing along.

After reading through the article, it is clear that no assumption is made on the coprimality of the $a_i$. This is never explicitly stated, but their many examples clearly show that they do not assume this, viz:

  • In the first paragraph of page 56, we have "We say that the coin types are multiplicative if every $a_m$ is a whole multiple of $a_{m-1}$. The coin types $1, 2, 10, 50, 100$ are therefore multiplicative, whereas not $1, 2, 5, 10, 25$. A special case of the multiplicative coin types are the binary, i.e. $1, 2, 4, 8, 16, \dots$"

  • In the first paragraph of page 57, $d_m$ is defined as the least common multiple of $a_1, a_2, \dots, a_m$. If they assumed all $a_i$ are pairwise co-prime, then $d_m$ could of course just be defined as their product instead.

But most importantly: in the examplesense that no added assumptions appear to be stated on page 61, in an example demonstrating the inequality you want, they use the coins $1,2,5,10,25,50$ and $100$ with $n = 10000$$a_i$ before the statement of the theorem.

As an aside, the Swedish used in this article is absolutely gorgeous, and is well worth the read. My only concern is that the authors assume that whoever is reading the article knows the Swedish currency system well, but not everyone is aware that there are $100$ öre in $1$ krona... to further complicate matters for modern readers, the öre (equivalent to a cent) is no longer in use in Sweden.

The article is [Blom, G. and Fröberg, C-E., Om Myntväxling, Nordisk Matematisk Tidskrift, 1962, Vol. 10, No. 1/2 (1962), pp. 55-69] for anyone who wishes to sing along.

After reading through the article, it is clear that no assumption is made on the coprimality of the $a_i$. This is never explicitly stated, but their many examples clearly show that they do not assume this, viz:

  • In the first paragraph of page 56, we have "We say that the coin types are multiplicative if every $a_m$ is a whole multiple of $a_{m-1}$. The coin types $1, 2, 10, 50, 100$ are therefore multiplicative, whereas not $1, 2, 5, 10, 25$. A special case of the multiplicative coin types are the binary, i.e. $1, 2, 4, 8, 16, \dots$"

  • In the first paragraph of page 57, $d_m$ is defined as the least common multiple of $a_1, a_2, \dots, a_m$. If they assumed all $a_i$ are pairwise co-prime, then $d_m$ could of course just be defined as their product instead.

But most importantly: in the example on page 61, in an example demonstrating the inequality you want, they use the coins $1,2,5,10,25,50$ and $100$ with $n = 10000$.

As an aside, the Swedish used in this article is absolutely gorgeous, and is well worth the read. My only concern is that the authors assume that whoever is reading the article knows the Swedish currency system well, but not everyone is aware that there are $100$ öre in $1$ krona... to further complicate matters for modern readers, the öre (equivalent to a cent) is no longer in use in Sweden.

The article is [Blom, G. and Fröberg, C-E., Om Myntväxling, Nordisk Matematisk Tidskrift, 1962, Vol. 10, No. 1/2 (1962), pp. 55-69] for anyone who wishes to sing along.

After reading through the article, no assumption is made on the coprimality of the $a_i$, in the sense that no added assumptions appear to be stated on the $a_i$ before the statement of the theorem.

As an aside, the Swedish used in this article is absolutely gorgeous, and is well worth the read. My only concern is that the authors assume that whoever is reading the article knows the Swedish currency system well, but not everyone is aware that there are $100$ öre in $1$ krona... to further complicate matters for modern readers, the öre (equivalent to a cent) is no longer in use in Sweden.

Source Link
Carl-Fredrik Nyberg Brodda
Carl-Fredrik Nyberg Brodda
  • 6.5k
  • 4
  • 27
  • 62

The article is [Blom, G. and Fröberg, C-E., Om Myntväxling, Nordisk Matematisk Tidskrift, 1962, Vol. 10, No. 1/2 (1962), pp. 55-69] for anyone who wishes to sing along.

After reading through the article, it is clear that no assumption is made on the coprimality of the $a_i$. This is never explicitly stated, but their many examples clearly show that they do not assume this, viz:

  • In the first paragraph of page 56, we have "We say that the coin types are multiplicative if every $a_m$ is a whole multiple of $a_{m-1}$. The coin types $1, 2, 10, 50, 100$ are therefore multiplicative, whereas not $1, 2, 5, 10, 25$. A special case of the multiplicative coin types are the binary, i.e. $1, 2, 4, 8, 16, \dots$"

  • In the first paragraph of page 57, $d_m$ is defined as the least common multiple of $a_1, a_2, \dots, a_m$. If they assumed all $a_i$ are pairwise co-prime, then $d_m$ could of course just be defined as their product instead.

But most importantly: in the example on page 61, in an example demonstrating the inequality you want, they use the coins $1,2,5,10,25,50$ and $100$ with $n = 10000$.

As an aside, the Swedish used in this article is absolutely gorgeous, and is well worth the read. My only concern is that the authors assume that whoever is reading the article knows the Swedish currency system well, but not everyone is aware that there are $100$ öre in $1$ krona... to further complicate matters for modern readers, the öre (equivalent to a cent) is no longer in use in Sweden.