a) Find all of the numbers between 1 and 1,000 that have 2 as their only prime factor.
b) What is the next number after 1,000 that had 2 as its only prime fractor?
a)
The only such numbers are the powers of 2:
2¹ = 2
2² =4
2³ = 8
2⁴ = 16,
2⁵ = 32
2⁶ = 64
2⁷ = 128
2⁸ = 256
and
2⁹ = 512
All other numbers are either non-prime or, if prime, are not divisible by 2.
b)
2¹⁰ =1024
Because all the numbers between 1 and 1000 that have 2 as their only prime
(a) that would be the powers of 2: 1,2,4,...
(b) 1024 = 2^10
a) To find all the numbers between 1 and 1,000 that have 2 as their only prime factor, we need to identify the numbers that can be expressed as 2 raised to some positive integer power.
One way to solve this is by repeatedly multiplying 2 by itself until we reach a number greater than 1,000. We start with 2 and continue multiplying by 2 until the result is greater than 1,000.
Here's a step-by-step process:
1. Start with the number 2.
2. Multiply 2 by itself to get 4.
3. Check if 4 is less than or equal to 1,000. If yes, it has 2 as its only prime factor, so we include it in the list. If not, proceed to the next step.
4. Multiply the current number (4) by 2 to get 8.
5. Repeat the process of checking if the current number is less than or equal to 1,000. If yes, include it in the list. If not, proceed to the next step.
6. Continue multiplying the current number by 2 and checking if it is less than or equal to 1,000 until reaching a number greater than 1,000.
By following this process, we can find all the numbers between 1 and 1,000 that have 2 as their only prime factor.
b) To find the next number after 1,000 that has 2 as its only prime factor, we can continue the process from part a) but start with the number 1,001 instead of 2.
Here's a step-by-step process:
1. Start with the number 1,001.
2. Multiply 1,001 by 2 to get 2,002.
3. Repeat the process of multiplying the current number by 2 until we reach a number that has 2 as its only prime factor.
By following this process, we will eventually find the next number after 1,000 that has 2 as its only prime factor.