Can anyone solve these. tell me what you did as well.?

Question

Mr. Byrd can buy a 757-page Algebra book for $14.99, or he can make copies of it for 5¢ a page. In either case, a 9% sales tax applies. 'What is the number of dollars in the positive difference between the costs for buying and copying the book? Express your answer to the nearest cent.

A positive integer has all single-digit prime factors. None of its prime factors occurs more than twice in the prime factorization, but the number is not a perfect square. What is the greatest possible number with these properties?

The numbers 3, 5 and 7 are important in many aspects of Japanese life. Most positive integers can be expressed as a sum of only 3's, 5's and 7's. For example, 15 can be expressed as 5 + 5 + 5 or 3 + 5 + 7. What is the greatest even integer that cannot be expressed as a sum of 3's, 5's and 7's?

Answer 1

1) Buying the book costs $14.99 plus 9%, so it's 14.99 + 0.09*14.99 = 1.09*14.99 = $16.34 (rounded to the nearest cent). Copying the book is 5 cents per page, or in dollars: 0.05*757, plus the 9% tax makes it 1.09*(0.05*757). Calculate this to the nearest cent, then take the difference from $16.34.

2) The only single-digit prime factors are 2, 3, 5, and 7 (and no, 1 is NOT a prime number). None of these repeast "more than twice". But taking (2*3*5*7) * (2*3*5*7) would be a perfect square. But if we drop one of the 2s, we get the greatest possible number that isn't a perfect square. So calculate what this would be.

3) First of all, there are the even integers 2 and 4 which won't work (6 is 3+3 and 8 is 3+5 ). Let's look at numbers that are 10 or greater:
Any number that ends in 0 is a multiple of 5, so it can be expressed as the sum of so many 5s. If it ends in 8, then it's some number ending in 0 plus 8, so you can write it as some multiple of fives plus 3+5=8. If it ends in 2, then you can write it as some number ending in 0 plus 5+7=12. And if it ends in 4, you can write it as some number ending in 0 plus 14, or 7+7. Finally, if it ends in 6, then you can write it as a multiple of ten plus 16, or 3+5+3+5.
So since any number greater than 10 cane be written this way, it looks like the highest even number that won't work is 4.

Answer 2

14.99 + 9%(14.99) = 14.99(1.09) = 16.34
757(0.05)(1.09) = 41.26
difference is 24.92

if each prime factor 2,3,5,7 is used twice, you get the perfect square 2²•3²•5²•7² = 44,100. to reduce it minimally take out 1 factor of 2 and get 22,050.

If M is even and M > 6 and M = 7a + 5b + 3c, then we can get M + 2, the next greater even, by adding another 7 and subtracting a 5, or by adding another 5 and subtracting a 3. So the greatest positive even integer that cannot be expressed this way is 4.