Tip #10 to Get a Top ACT Math Score

Example Problems

Easy

1. Which of the following lists the factors of 60 ?
   1. 1, 10, 20, 30, and 60
   2. 2, 4, 6, 10, 12, 20, 30, and 60
   3. 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60
   4. 1, 2, 3, 4, 5, 6, and 10
   5. 60, 120, 180, and 240
2. Which of the following is an odd number that is a factor of 140 ?
   6. 2
   7. 13
   8. 35
   10. 37
   11. 51

Medium

3. How many prime integer factors does the number 50 have?
   1. None
   2. 1
   3. 2
   4. 6
   5. 10

Hard

4. If a and b are positive integers greater than 1, such that the greatest common factor of a³b and a²b² is 36, then which of the following could a equal?
   6. 36
   7. 18
   8. 9
   10. 6
   11. 3
5. If a, b, and c are different odd prime numbers, how many factors does 2abc have?
   1. 4
   2. 5
   3. 15
   4. 16
   5. 24

Answers

1. C List the factors of 60, or better yet "Use the Answers." Just divide 60 by the numbers in each answer choice. If the numbers go into 60 evenly (without a remainder), then they are factors. We want the most complete list, the longest list that works.
2. H This question tests if you know the terms "odd" and "factor." Once you do, this is a simple "Use the Answers" type of question. Try each answer. Choice A does not work because 2 is not an odd number. Then, to test each of the other choices, simply divide 140 by the number. If it goes in evenly, which means you get an integer (no decimal or fraction), then it is a factor. For example, 140 ÷ 55 = 2.54, and therefore 55 is not a factor of 140. Choice H is correct since 140 ÷ 35 = 4.
3. C This question simply asks you to factor 50 and count how many of the factors happen to be prime numbers. To factor a number, make a list of pairs, as shown below.



When you reach 5 × 10, you know that you have them all since nothing between them multiplies to 50. Finding factors systematically like this is far better than randomly jotting down numbers. This is true for any ACT math question. Systematic and organized is better than scattered and random. It avoids careless errors, allows you to look back at your work, and helps you make the leap to the next step needed on a complicated question. So the prime numbers of our list are 2 and 5. Remember that 1 is not a prime number and that 2 is the only even prime number.

4. K "Use the Answers" is such a great strategy. Even for this "hard" question, just use the answers. Try each answer choice for a, and see which one could work. Choices F, G, H, and J, when plugged in for a, produce too large a number and would allow a number larger than 36 to be the greatest common factor. For example, when we try the number 9 for a, we get 9³b and 9²b². The greatest common factor of these numbers will be 81b, which is greater than 36. Plugging 3 in for a, we get 27b and 9b², which could have a greatest common factor of 36.
5. D To make this question less theoretical, just try different prime numbers for a, b, and c. We call this "Make It Real;" I'll discuss it more in Skill 18. Let's say a = 3, b = 5, and c = 7. Then 2abc = (2)(3)(5)(7) = 210. The long way to do this question is to list the factors of 210: 1, 2, 3, 5, 7, 6, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210. The shortcut is to notice that since 2, a, b, and c are prime, the factors will be 1, 2, a, b, c, 2a, 2b, 2c, ab, ac, bc, 2ab, 2ac, 2bc, abc, and 2abc.

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