Number Properties and Divisibility

0:00 So you're gearing up for the ASVAB, and maybe feeling a little, I don't know, anxious about the math part? Take a deep breath because you are in exactly the right place. Today, we are building a completely calculator free toolkit for ASVAB math by mastering foundational number properties. Now, the ASVAB doesn't allow calculators, which means we have to rely entirely on mental math and a solid sense of how numbers work to breeze through these questions. But don't worry, by the end of this explainer, you're going to have the mental shortcuts and the confidence you need to tackle these problems with total ease. Here's our roadmap for today. We'll start with basics and divisibility rules, move into factoring, GCF and LCM, and then cover integers, remainders, and absolute value. After that, we'll look at worked examples in action, point out common number property traps, and finish up with a check yourself section. All right, let's jump right into section one, basics and divisibility rules. First things first, we need to completely clear up the difference between a factor and a multiple. It's absolutely crucial. A factor divides a number perfectly, leaving zero remainder. Think of factors as the limited ingredients that make up a recipe.

1:01 Because of this, a number has a finite set list of factors. For instance, the factors of 12 are just one, two, three, four, six, and 12. That's it. The list stops there. On the flip side, a multiple is what you get when you multiply a number by any whole number. Multiples build a number up endlessly. It's like counting by sevens, seven, fourteen, twenty-one, twenty-eight, and so on forever. So simply put, factors break things down into a limited list and multiples build them up forever. Okay, now, what about prime versus composite numbers? A prime number is any whole number strictly greater than one that has exactly two factors, one and itself. Think of numbers like two, three, five, seven, and eleven. A composite number, however, has more than two factors. Take twenty-seven, for example. It can be broken down into three times nine. And here is a huge detail for the ASVAB that you should lock into your memory right now. The number one is neither prime nor composite.

2:02 And the number two is our one and only even prime number. Now, let's talk about divisibility rules. These are going to literally be your best friends because they completely bypass the need for long division. For two, five and ten, all you do is look at the very last digit. If the number ends in an even digit, it's divisible by two. If it ends in zero or five, it's divisible by five. And if it ends in a zero, it's divisible by ten. Super simple, right? Just check the tail end. Now, for three and nine, we have a different trick. You just add the digits together. If the sum of those digits is divisible by three, the entire massive number is divisible by three. And guess what? The exact same rule applies for nine. Let's build on that with a few slightly trickier rules that are absolute time savers. For four, look at just the last two digits to see if they form a multiple of four. For eight, you check the last three digits. For six, you just combine the tests we already know. The number has to pass the test for two, meaning it's even. And it has to pass the test for three.

3:02 Finally, the test for eleven is really cool. So take your time with this one. Starting from the left, you alternately subtract and add the digits. If your final result is zero or a multiple of eleven, then the whole thing is divisible by eleven. Moving right along, here is section two, factoring, GCF and LCM. Did you know every single whole number above one has a completely unique prime factorization? It's literally like a mathematical fingerprint. Prime factorization just means writing a number as a product of primes only. Take the number 90. If we start breaking it apart, we might say 90 is nine times 10. But neither nine nor 10 are prime, so we can't stop there. We keep splitting. The nine becomes three times three. The 10 becomes two times five. Put it all together, and the true prime factorization of 90 is two times three squared times five. So why do we care about those prime fingerprints? Well, we use them for our next two tools, the greatest common factor or GCF, and the least common multiple or LCM. The ASVAB loves to test if you know which one to pull out of your toolkit for a word problem.

4:07 Here's the secret. Use the GCF when a problem asks you to split things into the largest equal groups. Think about cutting different lengths of wood or ribbon into equal sizes with no waste. On the flip side, use the LCM when a problem involves repeating cycles that need to line up, like two different shuttle buses leaving a station at the exact same time again. Just remember it like this. GCF is for cutting down. LCM is for syncing up. Integers, remainders, and absolute value. Let's run through even and odd numbers really quickly. Keep in mind that an even number plus an even number is always even, and an odd plus an odd is also even. But a product, meaning multiplication, is only odd if every single factor is odd. If there is even one even number hiding in the mix, the whole product becomes even. Now, let me share a brilliant shortcut for consecutive integers. Those are numbers that follow each other exactly in order, like 18, 19, and 20. If you have three consecutive integers, their total sum is always exactly three times the middle number. So if a problem tells you three consecutive integers add up to 57, you just divide 57 by three to get 19.

5:12 That's your middle number. Boom. Your set is effortlessly 18, 19, and 20. All right, absolute value. Absolute value is simply a number's physical distance from zero on the number line. Because it's a distance, it is never ever negative. The huge mistake people make here is thinking you just instantly flip all subtraction signs into addition signs. You definitely don't. Absolute value bars act exactly like parentheses. You absolutely must simplify the math inside the bars first. For example, evaluate the absolute value of four minus 11. First, we do the math inside. Four minus 11 is negative seven. Then we take the absolute value of that result. The absolute value of negative seven is simply seven. All right, it's time to put it all together in section four, worked examples in action. Let's say you've got ribbons measuring 45 inches and 75 inches. You need to cut them into equal pieces as long as possible with no waste. That phrase cut into equal pieces is your secret trigger word.

6:12 It tells us right away we need the GCF. Step one, find the primes. 45 breaks down to nine times five, which is three squared times five. 75 breaks down to three times 25, which is three times five squared. To find the GCF, we take only the primes they share and we use the smaller power of each. They share a single three and a single five. So our GCF is three times five, which equals 15. Each piece will be exactly 15 inches long. One shuttle leaves every 15 minutes and another leaves every 20 minutes. Both leave right now. When do they next leave together? Repeating schedules lining up means we need our other tool, the LCM. Step one, 15 is three times five. 20 is four times five, which is two squared times five. For the LCM, we take every prime that appears across both numbers at its highest power. That means we take two squared times three times five. Multiply that all out and you get 60. No calculator required. The shuttles will leave together again in exactly 60 minutes. Let's tackle an absolute value example.

7:14 We need to evaluate the absolute value of two minus nine minus the absolute value of negative three. Let's do the inside of the first set of bars first. Two minus nine is negative seven. So our first term is the absolute value of negative seven, which simplifies to seven. Our second term is just the absolute value of negative three, which simplifies to three. Now we bring down that middle minus sign that was sitting between the two terms. Step three, seven minus three equals four. See how easy it is when you treat the bars exactly like parentheses? Almost there. Let's hit section five, common number property traps. Test makers absolutely love setting traps, but we're going to see right through them. First trap, calling one a prime number. Don't do it. It only has one factor and a prime number requires exactly two. Second, don't assume every odd number is prime. Look right at 21, 39, and 51. They are definitely composite. Third, as we just practiced, don't mix up GCF and LCM. Cutting into equal groups is GCF. Repeating schedules syncing up is LCM.

8:18 Fourth, don't drop the negative sign early on absolute value. You have to simplify the math inside the bars first. And finally, if a question asks for a remainder, don't give them the quotient. The remainder is just the leftover amount at the very end of your division. And it must always be smaller than the number you divided by. Finally, we've reached section six. Check yourself. You've built the toolkit. Now let's take it for a quick test drive. Which of the following numbers is prime? A 53, B 49, C 57, D 51. Take a second. Feel free to pause the explainer right here if you need a moment to think it through. All right, the correct answer is A 53. Why? Well, you want to test small primes to eliminate the fakes. 51 is actually three times 17. 57 is three times 19. And 49 is seven times seven. Because those have factors other than one and themselves, they are all composite. But no small prime divides evenly into 53, meaning its only factors are one and 53. So it's the real deal.

9:22 Let's try one more. Question two. Using the digit sum test we learned, which number can be divided evenly by nine? A 7521, B 2754, C 4105, D 6312. Again, go ahead and pause if you need to and carefully add up those digits for each option. And the correct answer is B 2754. If we add the digits of that option up, two plus seven plus five plus four, we get exactly 18. And since 18 is perfectly divisible by nine, that entire massive number 2754 is perfectly divisible by nine as well. And guess what? You just solved that without touching a calculator. You now have a powerful calculator free toolkit. When you're walking in to take the ASVAB, remember that the test isn't checking if you're a human calculator. It's just checking if you know the secret rules of the game. And now you definitely do. So what other impossible math barriers are you ready to break down next?