Quant

Question 1

Primes

Accepted Answer

"A prime is a natural number with exactly two distinct factors: 1 and itself". 5 is prime, because it has only two factors: 5 and 1. 12 is not prime, because it has more than two factors: 6, 4, 3, and 2 are all factors of 12.

Question 2

Factor Chart

Accepted Answer

he factor chart is a basic technique for finding all the factors (prime or non-prime) of any integer. This technique can also be useful for GMAT questions asking how many factors a particular integer has. The technique goes as follows: 1) Write the original integer on top of the table. 2) Divide the original Integer by the smallest possible factor: 1. Write "1" in the left column, and the result of the division in the right column. 3) Divide the original integer by the next higher factor: 2. Write "2" in the left column, and the result of the division in the right column. If the original integer is not divisible by 2, skip 2 and move on to higher possible factors: 3, 4, 5, etc. 4) Continue the process until the factors repeat themselves - stop when you're trying to divide the original integer by a factor that's already on the right side of the table. 5) You're done - the table now holds the factors (prime and not prime) of the original integer.

Question 3

Prime Tree

Accepted Answer

The technique goes as follows: 1) Divide the original integer by the smallest possible prime number: 2. If the integer is not divisible by 2, try dividing by the next smallest prime: 3, then 5, etc. Write the prime on the left side of the tree, result of the division on the right. 2) If the result is not prime - continue dividing by the smallest possible prime number, marking primes on the left, results on the right. 3) Repeat the process until the result of the last division is prime - then stop. 4) Circle the left-and-bottom-most integers on the tree - these are the prime factors of the original integer.

Question 4

Prime Tree

Accepted Answer

The technique goes as follows: 1) Divide the original integer by the smallest possible prime number: 2. If the integer is not divisible by 2, try dividing by the next smallest prime: 3, then 5, etc. Write the prime on the left side of the tree, result of the division on the right. 2) If the result is not prime - continue dividing by the smallest possible prime number, marking primes on the left, results on the right. 3) Repeat the process until the result of the last division is prime - then stop. 4) Circle the left-and-bottom-most integers on the tree - these are the prime factors of the original integer.

Question 5

Rules of divisibility

Accepted Answer

Divisible by 11 A number is divisible by 11 if the difference between the sum of its digits in the odd places and the sum of its digits in the even places is divisible by 11. In mathematical formula form: (sum of digits in odd places) - (sum of digits in even places) = a number that is divisible by 11. Let's look at 121,792 and see whether its divisible by 11. The sum of digits in 1st, 3rd and 5th places (odd places: 121,792) = 1+1+9 = 11 The sum of digits in 2nd, 4th and 6th places (even places: 121,792) = 2+7+2 = 11 The difference between these two sums is 11-11=0, which is divisible by 11 (0 divided by 11 is 0, which is an integer). Therefore the original number 121,792 is divisible by 11. Divisible by 10 For example, 480, 15720, 1,000,000,000 are all divisible by 10, because their last (units digit) is 0. Divisible by 9 Rule of divisibility by 9 - A number is divisible by 9 if the sum of its digits is divisible by 9. So 1,458 is divisible by 9, because 1+4+5+8=18, which is divisible by 9. Divisible by 8 a number is divisible by 8 if its last 3 digits form a 3-digit number that is divisible by 8. For example, 3,720 is divisible by 8, because 720 is divisible by 8. In the example above, 3,720 = 3,000 + 720. 3,000 is divisible by 1,000, which in turn is divisible by 8. So all we have to do is check and see whether the last 3 digits (720) are a number that is divisible by 8. Correct. The last 3 digits are 440, which is divisibile by 8 (If you're not sure, break down 440 into 400+40, both of which are divisibile by 8). Divisible by 7 The rule of divisibility for 7 is a little complicated, but fun: Multiply the number of hundreds the number has by 2. Add the remaining two digit number. If the result is divisible by 7, the original number is divisible by 7. 5040 is divisible by 7: There are 50 "hundreds" in 5040. Multiply 50 by 2 to get 100. Add the remaining two-digit number 40 to get 140. 140 is divisible by 7, so 5040 is divisible by 7 Divisible by 6 Simply put, a number is divisible by 6 if it is divisible by 2 and 3. So if a number clears the conditions set for divisibility by 2 (even last digit) and 3 (sum of digits divisible by 3), it is also divisible by 6. For example, 13,422 is divisible by 6, because it satisfies both conditions: a. Its units digit is even, so it's divisible by 2. and b. The sum of its digits (1+3+4+2+2 = 12) is divisible by 3, so it's also divisible by 3. Divisible by 5 Last digit 5 or 0 Divisible by 4 A number is divisible by 4 if its last two digits form a two-digit number that is divisible by 4. For example, 12560 is divisible by 4, because its last two digits form a number (60) that is divisible by 4. Divisible by 3 A number is divisible by 3 if the sum of its digits is divisible by 3. So 1455 is divisible by 3, because 1+4+5+5=15, which is divisible by 3. Divisible by 2 A number is divisible by 2 if its last digit (the unit's digit) is even. (Remember, 0 is an even number.)

Question 6

Adding and Subtracting Multiples

Accepted Answer

Now, for the general concept important for GMAT integer questions: Multiple of n ± multiple of n = MUST be a multiple of n. multiple of n ± NOT multiple of n = CANNOT be a multiple of n.

Question 7

Greatest Common divisor

Accepted Answer

Two non-zero integers have a common divisor if they are both divisible by the same positive integer. For example, 2 is a common divisor of 6 and 8, because both 6 and 8 are divisible by 2. All integers are divisible by 1, so 1 is always a common divisor of any two integers. However, some GMAT integers questions ask about the Greatest Common Divisor of two or more integers, and whether the G.C.D is greater than 1 or not. In the above case, 2 is also the Greatest Common Divisor of 6 and 8. Since prime numbers are divisible only by themselves and 1, any two primes will only have a G.C.D of 1. However, note that the reverse isn't necessarily true: The fact that two integers have a G.C.D of 1 does not necessarily indicate that they are Primes. This is a common misconception, easily disproved by example: 8 and 9 have a G.C.D of 1, even though neither of them is Prime. Common divisors - definition: Two non-zero integers have a common divisor if they are both divisible by the same positive integer. Remember this about the Greatest Common Divisor of two integers: Two primes will always have a G.C.D of 1. A G.C.D of 1 does not mean the two integers are prime - just that they have no common divisor greater than 1. Example: 8 and 9 have a G.C.D of 1. An integer can serve as the G.C.D of itself and another integer. Example: 6 and 12 have a G.C.D of 6.

Question 8

Integer question type involves fractions

Accepted Answer

A particular GMAT Integer question type involves fractions. Consider the following GMAT question: If y3 / 212 is an odd integer, which of the following could be the value of y?

GMAT (Fach) / Quant (Lektion)