I will post seven new problems every weekend, one for each day of the week. The idea is that you work on your own during the week and I will post answers the following Saturday. For those of you who practiced with me last year, the process is similar.
Week of Sept 25 – Oct 1: We are spending another week on counting divisors. Problems on this topic are modeled after those in previous tests and Introduction to Number Theory, the Art of Problem Solving.
I think some of the 7 new problems are HARD, and they rely on knowing how to apply the trick/formula that we practiced last week. Feel free to send me questions.
Click GoogleForm to submit answers by Oct 1.
Answers: 15, C, 123, 722, 49, 1 and 9, 16
Sept 25: Find the number of positive divisors of 324.
Sept 26. If the number of positive divisors of a positive integer n is odd, which of the following is always true?
A) n is odd B) n is even C) n is a perfect square D) n is a triangle number E) None of the above.
Sept 27: Find the number of distinct positive divisors of (30)^4, excluding 1 and (30)^4. (AMC)
Sept 28: What is the sum of the three numbers less than 1000 that have exactly five positive integer divisors? (MATHCOUNTS)
Sept 29: If n has 2 prime divisors and 9 total divisors, how many divisors does n^3 have?
Sept 30: If n has exactly 5 positive divisors, how many prime divisors does n have? How many positive divisors does n^2 have?
Oct 1: How many positive divisors do 480 and 840 have in common?
Week of Sept 18 – Sept 24: This week we continue with the topic on numbers and focus on counting divisors. Problems on this topic are modeled after those in previous tests and Introduction to Number Theory, the Art of Problem Solving.
Click GoogleForm to submit answers by Sept 24.
Prime factorization is a way of writing a positive integer as a product of its prime factors. For example 12 = 2^2 * 3; 19 = 19; 27 = 3^3; 300= 2^2*3*5^2. To count the number of positive divisors (or factors) a positive integer, it is often convenient to start with its prime factorization. If the prime factorization of n = p^i * q^j * r^k where p, q and r are primes and i, j and k are the exponents, the number of positive factors of n is (i+1)(j+1)(k+1). The formula also generalizes when the number of prime factors of n is not 3.
So the number of factors of 12 is (2+1)(1+1) = 6; the number of factors of 19 is 1+1 = 2; the number of factors of 27 is 3+1 = 4; the number of factors of 300 is (2+1)*(1+1)*(2+1) = 18. You may check your answers against the list of factors for each number.
Do you see why this formula works? Well, any factor of 12 has the form or 2^a*3^b. Note that the exponent a can take any value from 0, 1 and 2, and the exponent b can take any value 0 and 1. So the number of factors of 12 is 3*2 = 6.
See http://mathforum.org/library/drmath/view/57151.html for more explanation.
Answers: 2^4*3, 13^2, 2^4*3^1*5^2, 10, 3, 30, 8, 5, 24, 15, 16
Sept 18: Find the prime factorization of a) 48 b) 169 c) 1200
Sept 19: Using the formula above, find the number of positive factors of a) 48 b) 169 c) 1200.
Sept 20: How many of the positive factors of 48 are multiples of 2? You can list all even factors, or you may modify the formula. Hint: every even factor has to have at least one 2 in it. If an even number has prime factorization 2^i * p^j where p is a prime, then the number of even factors is i*(j+1).
Sept 21: How many of the positive factors of 48 are multiples of 3? Hint: use the same trick in counting the number of even factors.
Sept 22: How many of the positive factors of 1200 are multiples of 2?
Sept 23: How many of the positive factors of 1200 are multiples of 3?
Sept 24: How many of the positive factors of 1200 are multiples of 10? Hint: you need to extend the trick for counting the number of even factors.
Week of Sept 11 – Sept 17: This week we continue with the topic on numbers and focus on multiples and divisors. Problems on this topic are modeled after those in previous tests and Introduction to Number Theory, the Art of Problem Solving.
Click GoogleForm to submit answers.
Answers: 120, 420, 400, 156, 2, 12, 63, 16, 14, 4, 180, 21, 30
Sept 11: Find the least common multiple (lcm) of the following, a) lcm(24,30) b) lcm(35,84) c) lcm(50,80) d) lcm(52,78)
Sept 12: Find the greatest common divisor (gcd) of the following, a) gcd(14, 26) b) gcd(60,144) c) gcd(315,441) d) gcd(2000,2016)
Sept 13: How many of the first 100 positive integers are divisible by 7? (Mathcounts)
Sept 14: What is the result when we decrease gcd(6432,132) by 8? (AMC)
Sept 15: What is lcm(12,18,30)? (AMC)
Sept 16: Mr. A baked 252 cookies, Mr. B baked 105 cookies, Mr. P baked 168 cookies. Each baker packaged them with the same number of cookies in each package. What is the largest number of cookies that could be in each package? (Mathcounts)
Sept 17: How many positive integers less than 101 are multiples of 5 or 7, but not both at once?
Week of Sept 4 – Sept 10: This is the first week and we’ll start with simple problems on primes and composites. We will continue the topic over the next few weeks and you will see that the problems become harder gradually. Problems on this topic are either from previous tests or modeled after those in Introduction to Number Theory, the Art of Problem Solving.
Click GoogleForm to submit answers. They are due Sept 10 Sept 17 because of the beginning of the school year.
A prime number is a positive integer p>1 whose only positive divisors are 1 and p. The number 1 is not a prime, 2 is the only even prime, and the next few primes are 3, 5, 7, 11 …
Answers: 2, 3, 7, 17, 1, 0, prime, composite, composite, composite, 293, 73, 0, 2
Sept 4: Find the smallest prime divisor of a) 16 b) 57 c) 91 d) 289.
Sept 5: How many prime numbers are multiples of a) 7 b) 15?
Sept 6: Determine which of the following are prime and which are composites, a) 313 b) 343 c) 391 d) 427?
Sept 7: What is the largest prime between 200 and 300?
Sept 8: What is the largest 2-digit prime number whose digits are also each prime?
Sept 9: How many prime numbers are also perfect squares?
Sept 10: Find the remainder when the sum of the five smallest primes is divided by the sixth.