If 1 is added to every prime number less than 100, how many more of the resulting sums will be divisible by 4 than by 3? (If more of the sums are divisible by 3 than by 4, your answer will be negative.)
you can find a list of primes online
make the list , add one to each value , check the division
Thanks again Scott.
To solve this problem, we need to find the number of prime numbers less than 100 and then calculate how many of their sums are divisible by 4 and by 3.
Step 1: Finding the prime numbers less than 100
To find the prime numbers less than 100, we can use a prime number sieve or manually check each number from 2 to 99. After checking, we find that the prime numbers less than 100 are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
Step 2: Adding 1 to each prime number
We add 1 to each of the prime numbers:
3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98.
Step 3: Count how many sums are divisible by 4 and by 3
We check which of these numbers are divisible by 4 and by 3. If a number is divisible by 4, its remainder when divided by 4 will be 0. If a number is divisible by 3, the sum of its digits will be divisible by 3.
Counting the numbers divisible by 4, we find:
4, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98.
Counting the numbers divisible by 3, we calculate the sum of digits for each number and check if it's divisible by 3:
3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98.
From the above lists, we find that all the numbers from both lists are the same. Therefore, the number of sums divisible by 4 is the same as the number of sums divisible by 3.
So, the answer is zero.