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 May 14th, 2015, 08:39 AM #1 Newbie   Joined: May 2015 From: India Posts: 14 Thanks: 0 Math Focus: Number Theory Number of factors of a large number Find the number of factors of 10004000600040001
 May 14th, 2015, 08:49 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Your number is 73^4 * 137^4. The factors of the number are 73^m * 137^n with 0 <= m <= 4 and 0 <= n <= 4. How many are there in total?
May 14th, 2015, 05:07 PM   #3
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Hello, ishaanmj007!

Quote:
 $\text{Find the number of factors of }\,N \:=\:10,004, 000,600,040,001$

$N \;=\;10^{16}\,+\,4\cdot10^{12}\,+\,6\cdot10^8 \,+\,4\cdot10^4\,+\,1 \;=\;(10^4\,+\,1)^4$

$N \;=\;(10,001)^4 \;=\;(73\cdot137)^4 \;=\;73^4\cdot137^4$

$\text{If the prime factorization of }n\text{ is: }\,n \;=\;p^a\cdot q^b\cdot r^c\,\cdots$
$\;\;\;\text{the number of factors of }n\text{ is: }\:f(n) \;=\;(a+1)(b+1)(c+1)\,\cdots$

$\text{Therefore: }\:f(N) \:=\:(4+1)(4+1) \:=\:25.$

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