How many even factors does a number have? Now thats a rather odd question. Think about that.
Number Theory - Even Factors
Q.6: Number N = 2^{6} * 5^{5} * 7^{6} * 10^{7}; how many factors of N are even numbers?
1183
1200
1050
840
Correct Answer
Choice A. 1183.
Detailed Solution
The prime-factorization of 2^{6} * 5^{5} * 7^{6} * 10^{7} is 2^{13} * 5^{12} * 7^{6}
The total number of factors of N = 14 * 13 * 7
We need to find the total number of even factors. For this, let us find the total number of odd factors and then subtract this from the total number of factors. Any odd factor will have to be a combination of powers of only 5 and 7.
Total number of odd factors of 2^{13} * 5^{12} * 7^{6} = (12 + 1) * (6 + 1) = 13 * 7
Total number of factors = (13 + 1) * (12 + 1) * (6 + 1)
Total number of even factors = 14 * 13 * 7 - 13 * 7
Number of even factors = 13 * 13 * 7 = 1183
Alternative approach
Any factor of 2^{13} * 5^{12} * 7^{6} will be of the form 2^{p} * 5^{q} * 7^{r}.
Any even factor of 2^{13} * 5^{12} * 7^{6} will also be of the same form, except that p cannot be zero in this case.
p can take values 1, 2, 3, .........13 – 13 values
q can take values 0, 1, 2, .........12 – 13 values
r can take values 0, 1, 2, 3,….5 – 7 values
So, the total number of even factors be 13 * 13 * 7 = 1183
Correct Answer: 1183
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