What better way to spend the weekend than with your favorite friends: integers? Okay, maybe integers aren’t your best friends. But trust me, they are much better company than irrational numbers.

Below are five problems dealing with integers and integer properties. They are definitely not too challenging. In fact, many can be solved using mental math. That’s right: see if you can do any (or all) of the following problems in your head. Doing so will be a great way to see if your mental math skills are up to snuff. And, even if you miss a few, don’t worry—there will be scratch paper on the exam.

1. How many odd integers less than 100 are a multiple of 3 and a multiple of 5?

(A) 3

(B) 4

(C) 7

(D) 10

(E) 15

2. 5 times *n* is three less than twice *n*. What number is three less than twice *n*?

(A) -5

(B) -2

(C) -1

(D) 3

(E) 5

3. When p is divided by 7, the remainder is 4. When p is divided by 4, the remainder is 1. How many different values of p are less than 120?

(A) 2

(B) 4

(C) 5

(D) 11

(E) 18

4. How many factors of 6! are greater than 100?

(A) 4

(B) 5

(C) 7

(D) 10

(E) 12

5. The sum of seven consecutive integers is equal to 7. What is the average of the greatest and least terms in the series?

(A) 7

(B) 4

(C) 2

(D) 1

(E) -2

Answers:

1. A

2. A

3. B

4. B

5. D

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Hi Chris,

for the 4th question if the upper bound is not mentioned wont the answer be infinite

Can you explain the answer to question 3? i couldn’t figure that out 🙁

Thanks so much for this wonderful website. Would it be possible for you to explain the answer to number 4? Somehow I came up with 6 factors.

Henry thanks so much for the kudos, as well as your comment. It seems that I’ve become a victim of my own mental math (perhaps this calls for a special blog post on the dangers of not writing down each step.)

So, you’re right – the answer should be 6. That is there are six factors to 6!: 120, 144, 180, 240, 360, 720.

In general, the questions I post are my own, and unlike the Magoosh product questions, they have not been subjected to rigorous testing. (They usually pop into my head and then I write them down. Some have been tested on my students but not all.)

Again, my apologies to anyone out there who was wondering what happened to that sixth factor.