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Given the prime number 7, explain why every multiple of 7 who's factors do not share either 2,3, or 5 can be expressed by the following equations. $ 49 + 210k = y, 77 + 210k = y, 91 + 210k = y, 119 + 210k = y, 133 + 210k = y, 161 + 210k = y, 203 + 210k = y, $

Kevin Perez
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    Because the remainder, $\pmod{210}$ must be an element in $${1 \times 7, 2\times 7, 3\times 7, \cdots, 30\times 7.}.$$ Further, the prime factorization of this remainder, which is limited to primes less than $~\displaystyle \frac{210}{7} = 30,~$ must not include any element in the set ${2,3,5}.$ – user2661923 Jan 17 '23 at 21:16
  • @user2661923That makes perfect sense, but is the reason we can get other primes times 7, due to the fact that the remainder of that term will decompose into one of the aforementioned representations in the set? – Kevin Perez Jan 18 '23 at 04:06
  • The formulation is unlucky. What you mean are multiples of $7$ that have none of the prime factors $2,3,5$ , or alternative formulated , are coprime to $30$. By the way , you missed the form $210k+7$ – Peter Jan 18 '23 at 07:52
  • Yes, a number of the form $~n \times 7,~$ where $~n \in {2,3,\cdots,29},~$ must have a prime factorization that only involves primes less than $~30.~$ Further, by the constraint, none of the pertinent primes can be an element in $~{2,3,5}.$ – user2661923 Jan 18 '23 at 14:59

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