Combinatorics Question: How many positive integer factors does the number 457419312 = (2^4)*(3^5)*(7^6) have? The answer is 5*6*7=210. Can anyone explain why this is the answer? And, I found out that it's because (4+1)*(5+1)*(6+1) = 210. Why do we add one to each exponents of prime number factors?

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Combinatorics Question: How many positive integer factors does the number 457419312 = (2^4)*(3^5)*(7^6) have? The answer is 5*6*7=210. Can anyone explain why this is the answer? And, I found out that it's because (4+1)*(5+1)*(6+1) = 210. Why do we add one to each exponents of prime number factors?

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in any factor of this number, 2 can occur as a factor 0,1,2,3,4 number of times..ie..u can select any exponent of 2 in 5 ways..similarly u can select 3 in 6 ways (including 3^0) and so on...

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