Answer:  Clues 15 and 16 were incorrect.  So the number is not divisible by 16 and it’s not divisible by 17.   Bonus Answer:  $2^3\cdot 3^3\cdot  5^2\cdot 7\cdot 11\cdot 13\cdot 19\cdot 23\cdot 29\cdot 31=2,123,581,660,200$ Solution.  First note that if a number is not divisible by $n$, then it is not divisible by any multiple of $n$.  […]

Answer:  $\Large\frac{20^{24}}{24^{20}}$ is the bigger expression. To see this, we first look at the prime factorizations of the numerators and denominators in each expression: Let $A\Large = \frac{15^{16}}{16^{15}} = \frac{3^{16}\cdot 5^{16}}{2^{60}}=\frac{3^{36}\cdot 5^{16}}{2^{60}\cdot 3^{20}}$  and  $B\Large = \frac{20^{24}}{24^{20}}=\frac{2^{48}\cdot 5^{24}}{2^{60}\cdot 3^{20}}$  Now consider $\frac{20^{60}\cdot 3^{20}}{5^{16}}A = 3^{36}$  and  $\frac{20^{60}\cdot 3^{20}}{5^{16}}B = 24^8\cdot 5^8$. Observe that $3^{36}=3^{16}\cdot 3^{20}=9^8\cdot 3^{20}$. […]

*Most of the numbers can be created in multiple ways.  Below we provide a single solution for each.       26 = 3 * (2 + 6) + 2      27 = 6^2 – 3^2      28 = 6^2 – 2^3      29 = (6/2)^3 + 2      30 = (2 […]

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