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Number Theory Level 3

Find the number of (positive) factors of 10 ! 10! that are also multiples of 7 ! 7! , inclusive of 1 and itself. Notation : ! ! denotes the factorial notation. For example, 8 ! = 1 × 2 × 3 × ⋯ × 8 8! = 1\times2\times3\times\cdots\times8 .

32 27 30 31 29 28

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Rishik Jain by Rishik Jain , 21, India

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1 solution

10 ! = 7 ! × 8 × 9 × 10 \displaystyle 10!=7! \times 8\times 9\times 10 . We must check how many factors 720 720 has.

720 = 2 4 × 3 2 × 5 \displaystyle 720=2^4\times 3^2 \times 5 , So number of factors = ( 4 + 1 ) ( 2 + 1 ) ( 1 + 1 ) = 30 \displaystyle (4+1)(2+1)(1+1)=\boxed{30}

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