To find the highest power of 24 that divides 80, we analyze the prime factorization of 24 and 80.
- Prime factorization of 24 is 23×32^3\times 323×3.
- Prime factorization of 80 is 24×52^4\times 524×5.
The highest power of 24 dividing 80 depends on how many times 24=23×3124=2^3\times 3^124=23×31 can fit into 80. Since 80 has 242^424 but no factor of 3, it cannot be divided by 24 even once without remainder. Therefore, the highest power of 24 dividing 80 is zero. If the question is about the highest power of 24 dividing 80!80!80! (80 factorial), then:
- Calculate the maximum power of 2 in 80!80!80! using Legendre's formula:
⌊802⌋+⌊804⌋+⌊808⌋+⌊8016⌋+⌊8032⌋+⌊8064⌋=40+20+10+5+2+1=78\left\lfloor \frac{80}{2}\right\rfloor +\left\lfloor \frac{80}{4}\right\rfloor +\left\lfloor \frac{80}{8}\right\rfloor +\left\lfloor \frac{80}{16}\right\rfloor +\left\lfloor \frac{80}{32}\right\rfloor +\left\lfloor \frac{80}{64}\right\rfloor =40+20+10+5+2+1=78⌊280⌋+⌊480⌋+⌊880⌋+⌊1680⌋+⌊3280⌋+⌊6480⌋=40+20+10+5+2+1=78
- Calculate the maximum power of 3 in 80!80!80!:
⌊803⌋+⌊809⌋+⌊8027⌋+⌊8081⌋=26+8+2+0=36\left\lfloor \frac{80}{3}\right\rfloor +\left\lfloor \frac{80}{9}\right\rfloor +\left\lfloor \frac{80}{27}\right\rfloor +\left\lfloor \frac{80}{81}\right\rfloor =26+8+2+0=36⌊380⌋+⌊980⌋+⌊2780⌋+⌊8180⌋=26+8+2+0=36
Since 24 requires three 2's and one 3, the highest power of 24 dividing 80!80!80! is:
min(⌊783⌋,36)=min(26,36)=26\min\left(\left\lfloor \frac{78}{3}\right\rfloor,36\right)=\min(26,36)=26min(⌊378⌋,36)=min(26,36)=26
Thus, the highest power of 24 dividing 80!80!80! is 26. However, if the question is simply about the highest power of 24 dividing the number 80 itself, the answer is 0 because 24 does not divide 80 evenly even once. In summary:
- Highest power of 24 dividing 80 (the number) = 0.
- Highest power of 24 dividing 80!80!80! (factorial) = 26.
