4.j Number Patterns

Question 1

What is a terminating decimal?

Answer 1

A finite number of non-zero digits to the right of the decimal point.

Question 2

What is a non-terminating decimal?

Answer 2

Numbers that do not have a finite number of non-zero digits to the right of the decimal point. (infinite non-zero digits)

Question 3

Is 1/6 a terminating decimal?

Answer 3

No! It is non-terminating.

1/6 = 0.1666666666…..

Question 4

When will the decimal equivalent of a fraction be terminating?

Answer 4

The decimal equivalent will be terminating if the denominator (of the most reduced fraction), has a prime factorization containing only 2 and 5’s.

Question 5

Is 5/60 a terminating decimal?

Answer 5

5/60, reduce it down to 1/12

The prime factors of the denominator (12), is 2 x 2 x 3, since it contains a prime factor of 3 which is not 2 or 3, it must be non-terminating.

Question 6

Is 7/80 a terminating decimal?

Answer 6

7/80, can’t be reduced down!

The prime factors of the denominator (80), is 2 x 2 x 2 x 2 x 5, since it contains no prime factor other than 2 or 5, it must be terminating.

Question 7

When we divide consecutive positive integers by integer n, the pattern of remainders will be ….

Answer 7

When we divide consecutive positive integers by integer n, the pattern of remainders will be …. 0, 1, 2… n-1

Question 8

How would you answer this question?

Answer 8

Q/11

Thus, the remainder will follow a pattern from 0 to (n-1),
hence,
9, 10, 0, 1, 2, 3

Final answer is 3

Question 9

How would you answer this question?

Answer 9

Focus on spotting the pattern first

Question 10

How would you answer this question?

Answer 10

Spot pattern first

and know that 82*n must be even

Question 11

Which two integers remain the same regardless of what exponent they have?

Answer 11

0 and 1

0^n = 0
1^n = 1

Question 12

What is the pattern in the units digit of the positive powers of 2?

Answer 12

The units digit of the positive powers of 2 will follow the four-number pattern 2-4-8-6

Question 13

What is the pattern in the units digit of the positive powers of 3?

Answer 13

The units digit of the positive powers of 3 will follow the four-number pattern 3-9-7-1

Question 14

What is the pattern in the units digit of the positive powers of 4?

Answer 14

The units digit of the positive powers of 4 will follow the two-number pattern 4-6

Question 15

What is the pattern in the units digit of the positive powers of 5?

Answer 15

All positive integer powers of 5 end in 5

Question 16

What is the pattern in the units digit of the positive powers of 6?

Answer 16

All positive integer powers of 6 end in 6

Question 17

What is the pattern in the units digit of the positive powers of 7?

Answer 17

The units digit of the positive powers of 7 will follow the four-number pattern 7-9-3-1

Question 18

What is the pattern in the units digit of the positive powers of 8?

Answer 18

The units digit of the positive powers of 8 will follow the four-number pattern 8-4-2-6

Question 19

What is the pattern in the units digit of the positive powers of 9?

Answer 19

The units digit of the positive powers of 7 will follow the two-number pattern 9-1.

All positive odd powers of 9 end in 9.
All positive even powers of 9 end in 1.

Question 20

What is the pattern in the units digit of the positive powers of any number greater than 9?

Answer 20

The units-digit pattern of the powers of any integer greater than 9 has the same pattern as the powers of its units digit.

Eg. pattern of the positive powers of 12 –> same as that of 2

Question 21

How would you answer this?

Answer 21

Question 22

How would you answer this?

Answer 22

Question 23

How would you answer this?

Answer 23

Question 24

How would you answer this?

Answer 24

Question 25

When a whole number is divided by 10, the remainder is the _____________

Answer 25

When a whole number is divided by 10, the remainder is the UNITS DIGIT of the dividend

Question 26

When a whole number is divided by 100, the remainder will be the _____________

Answer 26

When a whole number is divided by 100, the remainder will be the LAST TWO DIGITS of the dividend

Question 27

How would you answer this?

Answer 27

Question 28

What is the special pattern when integers are divided by 5?

Answer 28

When integers with the same units digit are divided by 5, the remainder is constant. eg. 9/5 has a remainder of 4, so does 19/5, so does 59/5... etc

Question 29

How would you answer this?

Answer 29

Question 30

What is an evenly spaced set of numbers?

Answer 30

A set in which the numbers increase by a fixed amount when written in numerical order

Question 31

Is this an evenly spaced set of numbers? (11, 22, 33, 44, 55, 66)

Answer 31

(11, 22, 33, 44, 55, 66) This IS an evenly spaced set of numbers, given the difference between each number is exactly 11.

Question 32

What are the three ways in which evenly spaced sets commonly appear on the GMAT?

Answer 32

1) Set of consecutive integers (including sets of even and odd integers only) 2) Set of consecutive multiples (2, 4, 6, 8) 3) Set of consecutive numbers with a given remainder when divided by some integer (eg. remainder 2 when divided by 6: 14, 20, 26, 32)

Question 33

Two consecutive integers will never share ______________

Answer 33

Two consecutive integers will never share the same prime factors.

Question 34

What must the GCF between two consecutive integers be?

Answer 34

The GCF between two consecutive integers is 1. GCF (n, n+1) = 1

Question 35

If x is an integer greater than 1, and a and b are non-negative integers, then x^a is divisible by x^b if....

Answer 35

If x is an integer greater than 1, and a and b are non-negative integers, then x^a is divisible by x^b if.... a is GREATER OR EQUAL to b

Question 36

How would you answer this?

Answer 36

If you prime factorize 768 you get: 3 x 2^8 Then to divide 768/2^x, you can consider it as: 3 x 2^8 / 2^x Hence, you know that x must be equal to or smaller than 8.