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*** Josh and Charlie discovered a better solution ***
A. 48
B. 810,810,000
C. 1.738 x 10^26 (not 2.010 x 10^26)
The quantity of factors for a number can be found by breaking it into its prime factors.
72 = 2 x 2 x 2 x 3 x 3 = 2^3 x 3^2.
2^3 = 8. Now 8 has 4 distinct factors {1, 2, 4, 8} (one more than the exponent)
3^2 = 9 and 9 has 3 distinct factors {1, 3, 9} (one more than the exponent)
By multiplying the two sets of factors together, you get all of the factors for 72. {(1, 2, 4, 8), (3, 6, 12, 24), (9, 18, 36, 72)}
By multiplying the quantity of distinct factors, you get 3 x 4 = 12 distinct factors.
A. Reversing this process, for a number with 10 distinct factors:
10 = 5 x 2
Using the lowest 2 primes {2, 3} and the factors of 10, the following equation results:
2^(5-1) x 3^(2-1)
= 2^4 x 3 = 48
So the answer to Part A is 48.
B. 1000 distinct factors:
1000 = 5 x 5 x 5 x 2 x 2 x 2
First 6 primes {2, 3, 5, 7, 11, 13)
2^(5 – 1) x 3^(5 – 1) x 5^(5 – 1) x 7^(2 – 1) x 11^(2 – 1) x 13^(2 – 1)
= 2^4 x 3^4 x 5^4 x 7 x 11 x 13
= 810,810,000
C. 1,000,000 distinct factors
1,000,000 = 5 x 5 x 5 x 5 x 5 x 5 x 2 x 2 x 2 x 2 x 2 x 2
First 12 primes {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37}
2^4 x 3^4 x 5^4 x 7^4 x 11^4 x 13^4 x 17 x 19 x 23 x 29 x 31 x 37
= 2320^4 x 247,110,827
= 200,961,610,708,938,459,249,870,000
Correct answer from Josh and Charlie redistributed the factoring (2^9 vice 2^4 x 37) to get:
= 173,804,636,288,811,640,432,320,000
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