Here are a few problems to keep your number sense sharp during the summer months.

What is the greatest positive three-digit integer that is divisible by 5, 7 and 9?

*We know that any three-digit number that is divisible by 5, 7 and 9, is divisible by 5 × 7 × 9 = 315. The largest three-digit multiple of 315 is 315 × 3 = 945.*

What is the greatest possible product of a pair of two-digit integers, composed of the digits 8, 6, 4 and 2 if each digit is used exactly once?

*Using each of the digits 8, 6, 4 and 2 exactly once to make a pair of two-digit integers, we will achieve the greatest product of these two integers if one number has a tens digit of 8 and the other has a tens digit of 6. So, our products are 84 × 62 = 5208 and 82 × 64 = 5248. The greatest product is 5248.*

A proper divisor of a number is a divisor of the number that is not the number itself. What is the smallest positive integer that is less than the sum of its positive proper divisors?

*Right away we can eliminate any prime number since its only factors are 1 and itself. The sums of the proper divisors of positive integers beginning with 4 are 4: 1 + 2 = 3, 6: 1 + 2 + 3 = 6, 8: 1 + 2 + 4 = 7, 9: 1 + 3 = 4, 10: 1 + 2 + 5 = 8, and 12: 1 + 2 + 3 + 4 + 6 = 16. So, 12 is the smallest positive integer that is less than the sum of its proper divisors.*

For how many positive four-digit integers is the sum of its digits equal to the product of its digits?

*The only four digits that have the same sum and product are1, 1, 2 and 4, since 1 + 1 + 2 + 4 = 1 × 1 × 2 × 4 = 8. There are 4!/2! = 4 × 3 = 12 positive four-digit integers containing these digits.*

**♦ Page 1 of the linked PDF contains PROBLEMS & SOLUTIONS.
Page 2 contains ONLY PROBLEMS. ♦ **