[66] Number of prime numbers in \dfrac{6^{12}\times 35^{28}\times 15^{16}}{14^{12}\times 21^{11}}is:

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Aptitude

TNPSC G4

10th CBSE Maths

Number of prime numbers in $\dfrac{6^{12}\times 35^{28}\times 15^{16}}{14^{12}\times 21^{11}}is:$

112

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Explanation:

$\dfrac{6^{12}\times 35^{28}\times 15^{16}}{14^{12}\times 21^{11}}=$ $\dfrac{\left(2\times 3\right)^{12}\times \left(5\times7\right)^{28}\times \left(3\times 5\right)^{16}}{\left(2\times 7\right)^{12}\times \left(3 \times 7\right)^{11}}$

$=\dfrac{2^{12}\times 3^{12}\times 5^{28}\times 7^{28}\times 3^{16}\times 5^{16}}{2^{12}\times 7^{12}\times 3^{11}\times 7^{11}}$

$=2^{\left(12-12\right)}\times 3^{\left(12+16-11\right)}\times5^{\left(28+16\right)}\times7^{\left(28-12-11\right)}$

$=2^{0} \times3^{17} \times5^{44} \times7^{-5} $

$=\dfrac{3^{17}\times 5^{44} }{7^{5}}$

Number of prime factors = 17 + 44 + 5 = 66.

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