Find the sum of positive integer divisors and the number of positive integer divisors of $2^5{3}^4{5}^3{7}^{3}13$ .

Step 1: Divide the given number into prime factors The given number is already written in its prime factorization form: $2^5\cdot 3^4\cdot 5^3\cdot 7^3\cdot 13^1$ . Step 2: Determine the number of divisors To find the number of divisors, we use the formula $(a+1)(b+1)(c+1)...$ , where $a,b,c,...$ are the powers of the prime factors of the given number. In our case, the number of divisors is: $(5+1)(4+1)(3+1)(3+1)(1+1)=6\cdot 5\cdot 4\cdot 4\cdot 2=960$ . Step 3: Calculate the sum of the divisors To find the sum of the divisors, we use the formula $\frac{(p^{a+1}-1)}{(p-1)}\frac{(q^{b+1}-1)}{(q-1)}\frac{(r^{c+1}-1)}{(r-1)}...$ , where $p,q,r,...$ are the prime factors of the given number and $a,b,c,...$ are their respective powers. Applying this formula for our number: Sum of divisors = $\frac{(2^{5+1}-1)}{(2-1)}\frac{(3^{4+1}-1)}{(3-1)}\frac{(5^{3+1}-1)}{(5-1)}\frac{(7^{3+1}-1)}{(7-1)}\frac{(13^{1+1}-1)}{(13-1)}$ Sum of divisors = $\frac{(2^6-1)}{1}\frac{(3^5-1)}{2}\frac{(5^4-1)}{4}\frac{(7^4-1)}{6}\frac{(13^2-1)}{12}=63\cdot 121\cdot 156\cdot 840\cdot 14=75,076,408,240$ . So, the sum of the positive integer divisors of $2^5\cdot 3^4\cdot 5^3\cdot 7^3\cdot 13$ is 75,076,408,240 and the number of positive integer divisors is 960.