UPSC CDS 2 2022 Math Paper

We know
Total number of factors of xpyq is (p+1)(q+1)
Number of odd factors of xpyqzr if x is even and y and z are odd then the factors are
1×(q+1)(r+1)
We have,
⇒N=126×38×53
⇒N=26×26×36×38×53
⇒N=212×314×53
Total number of factors:- (12+1)×(14+1)×(3+1)
Total number of factors:- 13×15×4
Total number of factors =780
Statement I: The number of odd factors of N is 60 .
Number of odd factors =1×(14+1)×(3+1)
Number of odd factors =1×15×4
A number of odd factors =60.
Hence, Statement I is correct.
Statement II: The number of even factors of N is 720 .
Number of even factors = Total number of factors − Odd factors
Number of even factors =780−60
Number of even factors =720
Hence, Statement II is correct.
∴ Both statements are correct.