Question

list all prime numbers and composite numbers between 30 to 70
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Answer 1

It is an easy exercise:

composite numbers up to 70 (in fact up to 120) are divisible by one of the following: 2, 3,5, 7. Proof: The smallest divisor d

of a number n (different from 1) must be less than or equal the square root of that number. Otherwise you would have: n=de>n−−√n−−√=n

And to identify a composite number it is good enough to find a prime number that divides it. 2, 3 5, and 7 are the primes less than square root of 70.

It is easy to single out the numbers divisible by 2, 3 or 5: Divisibility by 5 is decided by looking at the last digit: if it is 0 or 5, then your number is divisible be 5. For divisibility by 2 you single out the even numbers, and the divisibility by 3 you should have by heart (otherwise add the two digits, and mark those with sum of digits equal to 3, 6, 9, 12, 15 or 18.

Mathematicians combine the 2, 3 case by saying: look at all multiples of 6 = 2*3. All numbers greater 3 and not divisible by 2 or 3 are plus or minus 1 of a multiple of 6.

So your candidates are (besides 2 and 3):

(5, 7, 11, 13, 17, 19, 23, 25, 29, ) 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 65, 67,

Remove the ones with a last digit 5 (zero is no longer present):

31, , 37, 41, 43, 47, 49, 53, , 59, 61, , 67,

And now look at divisibility by 7. No trick needed, you know the multiples of 7 by heart. It is just 49, that needs to be removed:

31, , 37, 41, 43, 47, , 53, , 59, 61, , 67,

Here you go.

Answer 2

Answer:

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Step-by-step explanation:

PRIME NUMBERS;

31, 37, 41, 43, 47, 53, 59, 61 and 67.

COMPOSITE NUMBERS;

32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, and 69