You are asked to see how many values less than 5000 are multiples of 7. The first 6 numbers (1,2,3,4,5,6) can't be multiples of 7 so 7 is your first number. 5000 is not divisible by 7 so you got to find the last value that is a multiple of 7. I like to do this by dividing 5000 by 7 and then multiplying it by whatever number they gave me (if it's a whole number then that's a multiple, if it's a decimal then that means it is not a multiple). 5000/7= 714.28 so multiply 7 by 714 and you'll get 4998 as your answer. This is in fact a multiple of 7 and we can conclude that the numbers we are working with are in the range of 7-4998. From here I subtract 7 from 4998 and then divide it by 7 to see how many integers are multiples of 7. I came out with 714 possible values. From here it asks you to see which one of these multiples of 7 is also the product of 11 and an even number. The lowest possible value that could be is 11*2= 22. So look at the multiples of 22 until you see one that is a multiple of 7. It took 7 tries for me to realize that the seventh multiple of 22 is also a multiple of 7 (154). Find the range again by the same technique used to find the range for 7 and you get a range of possible values from 22-4994. Again subtract 22 from 4994 and you get 226+1 for a total of 227 multiples of 22. From here we need to find out which multiples of 22 are multiples of 7. Again our range is now from 7 to 224 so subtract 7 from 224 and you get 217. Divide that value by 7 and you get 31 values. Don't forget to add +1 and your final answer is 32!. We add +1 to include the last value of the range. The way i like to think about it is asking yourself "How many integers are in 1-10?" Well obviously 10 but if we subtract 1 from 10 our answer is 9 so don't forget to include the last value of the set to get that total of 10! Sorry for making this so long, but i wanted to be as descriptive as possible. Cheers!

I saw this as a patterns and sequences type problem. I found that there are 14 multiples of 7 between 0-100. 7 of those multiples are even numbers. 100-200, 200-300, and 300-400 do as well. Then when we get to 400-500 we need to worry about going over 5,000. 11*406, 11*420, 11*434 and 11*448 all stay under. So we end up with 4 groups of 7 plus the 4 from the last group.

the positive integers must contain at least 7,11 and 2 primes thus, 7*11*2 =154 will be the first integer that satisfies the mentioned conditions.
and the subsequent positive integers are n*154, where, n=1,2,3,4,5,6....... up to 32. Because 32*154= 4928 the highest integer less than 5000.
So the answer is 32.
Alternative way: just divide 5000 by 154, 5000/154=32.47, therefore, 32 is the answer

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