Numbers and Divisibility

$x^{100}+2x^{99}+k$ is exactly divisible by $(x+1)$

$321729$not divisible by $2,3,6$

Required numbers are $10,15,20,25,......96$
This is an A.P. in which $a=10,d=5$ and $l=95$
${t} _{n}=95$ $\Rightarrow$ $a+(n-1)d=95$
$\Rightarrow$ $10+(n-1)\times 5=95$
$\Rightarrow$ $(n-1)\times 5=85$
$\Rightarrow$ $(n-1)=17$
$\Rightarrow$ $n=18$
$\therefore$ Required Sum $=\cfrac { n }{ 2 } (a+l)=\cfrac { 18 }{ 2 } \times (10+95)=(95\times 105)=945$

Required numbers are $24,30,36,40,.....96$
This is an A.P. in which $a=24,d=6,l=96$
Let the number of terms in it be $n$.
Then ${t} _{n}=96$ $\Rightarrow$ $a+(n-1)d=96$
$\Rightarrow$ $24+(n-1)\times 6=96$
$\Rightarrow$ $(n-1)=12$
$\Rightarrow$ $n=13$
Required number of numbers $=13$

The smallest positive integer that is divisible by 2, 3, and 5 is the least common multiple (LCM) of 2, 3, and 5. The prime factorization of 2 is (2), the prime factorization of 3 is (3), and the prime factorization of 5 is (5). Therefore, the LCM of 2, 3, and 5 is (2 \cdot 3 \cdot 5 = 30). Therefore, the smallest positive integer that is divisible by 2, 3, and 5 is 30.

There are 33 positive integers less than 100 that are divisible by 3 or 5.

The GCD of two numbers is the largest positive integer that divides both numbers without leaving a remainder. We can find the GCD of 12 and 18 by using the Euclidean algorithm: 18 = 12 * 1 + 6, 12 = 6 * 2 + 0. Therefore, the GCD of 12 and 18 is 6.

The smallest positive integer that is divisible by 2, 3, and 5 is 30.

To find the greatest common divisor (GCD) of 12 and 18, we can use the Euclidean algorithm. We divide 18 by 12 to get a quotient of 1 and a remainder of 6. Then we divide 12 by 6 to get a quotient of 2 and a remainder of 0. Therefore, the GCD of 12 and 18 is 6.

The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder. The GCD of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 without leaving a remainder.

The sum of all positive integers less than 100 that are divisible by 3 is (3 + 6 + 9 + ... + 96 + 99) = 1665. The sum of all positive integers less than 100 that are divisible by 5 is (5 + 10 + 15 + ... + 90 + 95) = 1015. However, we have counted the numbers that are divisible by both 3 and 5 twice, so we need to subtract them once. The numbers that are divisible by both 3 and 5 are (15, 30, 45, 60, 75, 90), and their sum is 315. Therefore, the sum of all positive integers less than 100 that are divisible by 3 or 5 is 1665 + 1015 - 315 = 2365.

The sum of all the elements of $S$ is equal to the sum of the multiples of 3 less than 100 plus the sum of the multiples of 5 less than 100 minus the sum of the multiples of 15 less than 100. The sum of the multiples of 3 less than 100 is 3 + 6 + 9 + ... + 96 + 99 = 1683. The sum of the multiples of 5 less than 100 is 5 + 10 + 15 + ... + 90 + 95 = 2223. The sum of the multiples of 15 less than 100 is 15 + 30 + 45 + ... + 75 + 90 = 360. Therefore, the sum of all the elements of $S$ is 1683 + 2223 - 360 = 1863.

The sum of all the elements of $S$ is equal to the sum of the multiples of 3 less than 100 plus the sum of the multiples of 5 less than 100 minus the sum of the multiples of 15 less than 100. The sum of the multiples of 3 less than 100 is $3 + 6 + 9 + ... + 96 + 99 = 1683$. The sum of the multiples of 5 less than 100 is $5 + 10 + 15 + ... + 90 + 95 = 2223$. The sum of the multiples of 15 less than 100 is $15 + 30 + 45 + ... + 75 + 90 = 360$. Therefore, the sum of all the elements of $S$ is $1683 + 2223 - 360 = 1863$.

A number is divisible by 5 if its last digit is either 0 or 5. In this case, 345 is the only number with a last digit of 5, so it is divisible by 5.

The greatest common divisor (GCD) of two integers is the largest positive integer that divides both integers without leaving a remainder. The GCD of 18 and 24 is 6 because 6 is the largest positive integer that divides both 18 and 24 without leaving a remainder.