Recall that the number n factorial, which is written n!, is equal to the product
n! = 1 • 2 • 3 • • • (n − 1) • n.
(a) Find the highest power of 2 dividing each of the numbers 1!, 2!, 3!, . . . , 10!.
(b) Formulate a rule that gives the highest power of 2 dividing n!. Use your rule to compute the highest power of 2 dividing 100! and 1000!.
(c) Prove that your rule in (b) is correct.
(d) Repeat (a), (b), and (c), but this time for the largest power of 3 dividing n!.
(e) Try to formulate a general rule for the highest power of a prime p that divides n!. Use your rule to find the highest power of 7 dividing 1000! and the highest power of 11 dividing 5000!.
(f) Using your rule from (e) or some other method, prove that if p is prime and if pm divides n! then m < n/(p − 1). (This inequality is very important in many areas of advanced number theory.)
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