14. Prove that 7 divides n = 10m+k if and only if 7 divides m -...

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14. Prove that 7 divides n = 10m+k if and only if 7 divides m - 2k. (I would try to multiply 10m + k by something and add or subtract a multiple of m-2k in such a way that the result is clearly 7x something. Then, if 7 divides one part of it, it must clearly divide the other part. You just have to make sure that what you multiply each part by is not also a multiple of 7.) 15. Prove that 19 divides n = 10m + k if and only if 19 divides m + 2k. Use the rule to show 19 | 102239 and 19 31563214. 14. Prove that 7 divides n = 10m+k if and only if 7 divides m - 2k. (I would try to multiply 10m + k by something and add or subtract a multiple of m-2k in such a way that the result is clearly 7x something. Then, if 7 divides one part of it, it must clearly divide the other part. You just have to make sure that what you multiply each part by is not also a multiple of 7.) 15. Prove that 19 divides n = 10m + k if and only if 19 divides m + 2k. Use the rule to show 19 | 102239 and 19 31563214.