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Let k be the largest integer such that 343!/7^k is an integer. What is the remainder when 343!/7^k is divided by 7?

sudsw12 Jan 5, 2019

#1**+3 **

Factor 343!

2^337×3^169×5^83×7^57×11^33×13^28×17^21×19^18×23^14×29^11×31^11×37^9×41^8×43^7×47^7×53^6×59^5×61^5×67^5×71^4×73^4×79^4×83^4×89^3×97^3×101^3×103^3×107^3×109^3×113^3×127^2×131^2×137^2×139^2×149^2×151^2×157^2×163^2×167^2×173×179×181×191×193×197×199×211×223×227×229×233×239×241×251×257×263×269×271×277×281×283×293×307×311×313×317×331×337 (918 prime factors, 68 distinct).

**It follows from the above that: [343! / 7^57] is an integer. And:**

**[343! / 7^57] mod 7 = 6**

Guest Jan 6, 2019