Two Conditions Questions | Math Reasoning

Introduction

Two conditions questions are unique to the CLT among college admissions tests. They provide a range of integers and ask the student to determine how many integers in the given range meet both, at least one, or neither of two criteria. Unfamiliarity may make them seem challenging at first, but they need not be intimidating, especially those that appear in the first half of the CLT Quantitative Reasoning section. Try the question below, then explore strategies for these types of questions before attempting examples at whichever difficulty levels are appropriate for your current abilities.

Approach Question

Note any key terms that require defining in order to answer the question below. Then make your best attempt.

How many integers between $100$ and $500$ , inclusive, meet both of the following conditions given in the statements below?

1: The number is a perfect square.

2: The product of the number’s digits is not zero.

A) 0
B) 2
C) 9
D) 11

Explanation

Among the 401 numbers in the range from $100$ to $500$ inclusive, we need to find how many meet both listed criteria. Perfect squares are those values that are equal to integer squared (the first few examples are $1$ , $4$ , $9$ , and $16$ ). (We highly recommend you memorize all the perfect squares all the way up to $2 0^{2}$ ). We would do well to begin with this condition, because the other condition, having digits whose product is not zero, contains hundreds of examples in the range between $100$ and $500$ inclusive.

To list the perfect squares, we can start at the smallest number in the range, $100$ , noting that we have already found our first perfect square, since $100 = 1 0^{2}$ . From there, we can list $1 1^{2}$ , $1 2^{2}$ , etc., until we get close to $500$ . That gives us $100$ , $121$ , $144$ , $169$ , $196$ , $225$ , $256$ , $289$ , $324$ , $361$ , $400$ , $441$ , and $484$ . As you may notice, the difference between any two consecutive numbers is growing as we go, so the next number after $484$ will be too large to stay less than or equal to $500$ .

The list above contains 13 numbers. But we need to remove any number where the product of its digits is zero. That eliminates $100$ and $400$ . Eleven numbers remain, so the answer is D.

Definitions

Integer: An integer is a whole number–positive, negative, or zero. (Technically, whole numbers in math are only positive, but the informal definition of “whole number” will do here. So $3$ is an integer, but $3.1$ is not.)
Inclusive: The word inclusive means the numbers on the extremes of the list are included. So in this case, both $100$ and $500$ are included. To find the numbers of integers between $100$ and $500$ inclusive, we subtract the two numbers and add $1$ to the difference. There are $401$ (not $400$ !) numbers in this set.
Condition: A condition is a statement that must satisfied or agreed with in order for something to be achieved–in this case, the right answer!
Square and square root: A square is the result of multiplying a number by itself. So, $16$ is the square of $4$ . Note that a square root is the opposite of a square. $4$ is the square root of $16$ .
Product: A product is the result of multiplying two or more numbers.

Topics for Cross-Reference

Factors and Multiples

Variations

Instead of asking for integers that meet both given conditions, the CLT may ask you to meet at least one or neither of the two conditions.

Strategy Insights

To count inclusively, subtract the smallest number in the range from the largest and add one. To put it another way, if you know that the statistical quantity known as range is simply the largest number minus the smallest, then inclusive counting adds one to the range.
When dealing with two conditions in a question like this, always start with the condition that feels simpler to approach first. Usually, this will be the condition with fewer examples. This is called the “limiting condition.”

Flashcard Fodder

Important reminder: The CLT does not allow calculator usage! But as we can see from the approach question at the beginning of this module, the test still expects you to know your multiplication facts. We recommend that you memorize the times tables up to $15 \times 15$ . Using physical flashcards or a flashcard program like Quizlet is highly recommended. It’s important to be automatic with your multiplication facts so you can spend your time with actual problem-solving.
In addition, even beyond $15$ , you should memorize the perfect squares up to $20 \times 20$ . You’ll be happy you did when the question calls for such calculation and you’re not stuck writing out long multiplication!
Always make definition flashcards for anything listed under “key terms” above that you are not already familiar with.

Sample Questions

Difficulty 1

How many integers between $100$ and $200$ , inclusive, meet both of the following conditions given in the statements below?

1: The number is divisible by $13$ .

2: The number is even.

A) 4
B) 6
C) 8
D) 10

(spoiler)

The answer is A. Following the strategy of starting with the more accessible statement, let’s begin with statement 1 because it will have fewer examples than statement 2 - better not to list all the even numbers from $100$ to $200$ if we don’t have to! If you take our advice and do your times tables up to $15$ , you will be in good shape working with $13$ as a factor. But if you need help, consider that $13 \times 10 = 130$ . From $130$ , we can count up and down by $13$ at a time. The lowest number will be $104$ and the highest will be $195$ . Among this whole list, we now count the even integers: $104$ , $130$ , $156$ , and $182$ . There are four.

Difficulty 2

How many numbers between $10$ and $50$ (inclusive) meet both of the conditions given in the statements below?

1: The number is divisible by $4$ .

2: The last digit is a composite (non-prime), nonzero number.

A. 6
B. 7
C. 8
D. 9

(spoiler)

The answer is A. The simpler of the two tasks here is to list the multiples of $4$ in the relevant range: $12$ , $16$ , $20$ , $24$ , $28$ , $32$ , $36$ , $40$ , $44$ , $48$ . Which of these fits the second condition? We can rule out $20$ and $40$ because the last digit must be nonzero. But the last digit also must be composite, and we mustn’t forget that 2 is prime. That takes out $12$ and $32$ as well, leaving only 6 examples.

Difficulty 3

How many integers between $1$ and $100$ (inclusive) meet both of the conditions below?

1: The integer is a prime number doubled.

2: The integer’s units digit is $2$ .

A. 3
B. 4
C. 5
D. 6

(spoiler)

The answer is A. The second statement is the simpler, and it doesn’t take long to list out $2$ , $12$ , $22$ , etc. all the way to $92$ . Which of these is a prime number doubled? Certainly not $2$ , since $1$ isn’t prime. $12$ doesn’t work either since half of it is $6$ . In fact, all the numbers that are double an even number don’t work, so we can take out $32$ , $52$ , $72$ , and $92$ . Of the remaining numbers, $22$ looks good; half of it is $11$ , which is prime. $42$ might look good at first, but $21$ is not prime since it is divisible by $3$ . The remaining numbers, $62$ and $82$ , join $22$ as good candidates, so there are a total of 3.

We can sum up by noting that if a number’s units digit is $2$ , half that number must end in either $1$ or $6$ . All the values that end in $6$ , being even numbers, are immediately eliminated from prime number contention.

Difficulty 4

How many even integers between $2$ and $200$ (inclusive) meet at least one of the conditions below?

1: The integer is a perfect square.

2: The integer is a multiple of $98$ .

A. 8
B. 9
C. 14
D. 16

(spoiler)

The answer is A. From the second statement (and make sure you don’t confuse multiple with factor here) we find a very small number of fitting examples: just $98$ and $196$ . So we will need to list the examples filling the first statement. But let’s keep two important words/phrases in mind. First, unlike most “Two Conditions” questions, this question is asking which number meet at least one condition (not both). Second, we dare not overlook the word “even” in the question, which narrows down our list considerably.

With these things in mind, we start listing the perfect squares starting with $2^{2} = 4$ . How high does the list go? This is another example of a problem showing why knowing multiplication facts up to $15$ is helpful: $1 4^{2} = 196$ , so that’s our last example. So the numbers satisfying the first condition are $4$ , $16$ , $36$ , $64$ , $100$ , $144$ , and $196$ . Adding the two examples we got from the other statement, we get a total of 9 numbers. But watch out for double counting! $196$ is present in both lists but only gets counted once. Therefore the answer is A.

Difficulty 5

How many integers between $80$ and $120$ (inclusive) meet neither of the conditions below?

1: The integer is a multiple of $3$ .

2: The integer’s digits add up to an even number.

A. 10
B. 11
C. 12
D. 13

(spoiler)

The answer is C. Like the last question, this “Two Conditions” question is unusual, but this time it’s because the key word is “neither”. We must catalog all the numbers that don’t fulfill either condition.
To do this, we should list the non-multiples of 3 in the given range, even though the list is pretty long. (Try it now on your scratch paper.) Starting with $80$ and ending $119$ , we have a total of 26 integers. Now let’s go through those integers and cross out the ones whose digits add up to an even number. So in the $80$ ’s we can cross out $80$ , $82$ , $86$ , and $88$ . Continuing the process, we’ll get rid of $91$ , $95$ , and $97$ . The numbers greater than $100$ in the “cross-out” category are $101$ , $103$ , $107$ , $109$ , $112$ , and $116$ . There should be a total of 12 numbers left on your list.

Interestingly, if you scrutinize your list carefully, you’ll notice that exactly three numbers remain from each “decade” in our list: the 80’s, the 90’s, the 100’s, and the 110’s. At least we have some sort of pattern, so if you had little time on this question and had to make an educated guess, you might find three candidates in the 80’s and assume that each of the following decades has the same number of candidates. There are worse educated guesses out there, no doubt!

For Reflection

How will you approach “two conditions” questions on test day? Write down at least three takeaways from this module.
Rate the difficulty of these questions for you from 1 (no problem) to 5 (problem!). This will help you decide when to answer them and when to skip them on test day. Make sure to account for the time it takes to complete this sort of question. (Harder examples of this type may take too long to be worth it.)
Come up with a mnemonic (memory aid) to memorize a key strategy from this lesson. It could be something visual, like imagining perfect squares actually appearing as giant square shapes of perfection, or something with wordplay or rhyming like “Find the limits, that’ll keep you in it.” Don’t be afraid to choose something odd as a mnemonic: the stranger it is, the more likely you are to remember it!