4.e Divisibility

Question 1

What does “divisible by” mean?

Answer 1

If two numbers can divide into each other, without leaving any remainder

Question 2

How can you express a variable being divisible by another using fractions?

Answer 2

x is divisible by y if:

x / y = INTEGER

Question 3

How can you check for divisibility using prime factors?

Answer 3

x is divisible by y if:

the prime factorization of x contains the prime factorization of y

15 = 3 x 5
60 = 2 x 2 x 3 x 5
–> as 60 contains 3x5, which are the prime factors of 15, this is divisible!

Question 4

If y is a factor of x, and z is a factor of y, _____________

Answer 4

If y is a factor of x, and z is a factor of y, z is ALSO A FACTOR of x

Question 5

If any number is divisible by another number, what should you remember about the factors?

Answer 5

That the factors of the dividing number can also divide evenly into the number you are dividing

Question 6

If x is divisible by two or more numbers, it must ______________

Answer 6

If x is divisible by two or more numbers, it must ALSO be divisible by the LCM of those numbers

Question 7

How would you do this question:

Answer 7

Prime factorize 12 and 100

12 = 2 x 2 x 3
100 = 2 x 2 x 5 x 5

LCM = 2 x 2 x 3 x 5 x 5 = 300

As you know the LCM of any divisible integers must also divide into the original number. Therefore you know factually that 300 is a divisor of the number.

Question 8

If X and Y are positive integers, and X is divisible by j, and Y is divisible by k,, then..?

Answer 8

X is divisible by j
Y is divisible by k

Then, jk is the greatest integer that must evenly divide into XY

Question 9

What should you do when facing this fraction:

Answer 9

Make sure j/k is expressed in simplest terms.

Then, you know that N must be divisible by k

Question 10

What can you infer about the value of N?

Answer 10

Considering 15/17 is in its most reduced form, N must be divisible by 17

Question 11

How could you apply percentages to these divisibility concepts?

(pre-tax price of “W”, integer, if a 16% sale tax is added and the remaining fee is still an integer, what can you infer?)

Answer 11

Sales tax = 16% of original

T = 16/100 * W
T = 4/25 * W
T = 4W/25

So, W is divisible by 25
And, T is divisible by 4 (considering 25T/4 = W)

Question 12

How would you tackle this question?

Answer 12

WATCH OUT!

Don’t do 84/100, because its not really asking for that!
Instead, consider the percentage as 0.84 (technically the same as the percentage lol), and divide THAT by 9.

Question 13

Which numbers are divisible by 0?

Answer 13

No numbers

Question 14

Which numbers are divisible by 1?

Answer 14

1

Question 15

Which numbers are divisible by 2?

Answer 15

All numbers for which the units digit is even (0, 2, 4, 6, 8)

Question 16

Which numbers are divisible by 3?

Answer 16

If the sum of all the digits is divisible by 3

Question 17

Which numbers are divisible by 4?

Answer 17

If the last two digits are divisible by 4

(00 IS divisible by 4!)

Question 18

Which numbers are divisible by 5?

Answer 18

All numbers for which the units digit is a 0 or a 5

Question 19

Which numbers are divisible by 6?

Answer 19

If the number is even AND its digits sum to a multiple of 3

eg. 18
Is EVEN, and its digits add 1 + 8 = 9, which is a multiple of 3

Question 20

Which numbers are divisible by 7?

Answer 20

No logic here, must do full calculation

Question 21

Which numbers are divisible by 8?

Answer 21

Take the last three digits and divide by 8, if no remainder, it must be even.

Eg. 1160, 160/8 = 20
So 8 can divide into 1160

Question 22

Which numbers are divisible by 9?

Answer 22

If the sum of all its digits is divisible by 9

Question 23

Which numbers are divisible by 10?

Answer 23

If the units digit is a 0

Question 24

Which numbers are divisible by 11?

Answer 24

COMPLEX

The sum of the odd-numbered placed digits,
- MINUS
The sum of the even-numbered placed digits

remember its saying PLACED, so the units digits is the first number to the left, and is thus odd-placed.

-> THAT result must be a multiple of 11

Question 25

Which numbers are divisible by 12?

Answer 25

Any number that is divisible by both 3 and 4

Question 26

The product of any n consecutive integers must be divisible by ___

Answer 26

The product of any n consecutive integers must be divisible by n!

Question 27

What must 5 x 6 x 7 be divisible by?

Answer 27

5 x 6 x 7 are three consecutive numbers, so they MUST be divisible by 3!

Question 28

What must 20 x 21 x 22 x 23 be divisible by?

Answer 28

20 x 21 x 22 x 23 four consecutive numbers, must be divisible by 4! 4! = 4 x 3 x 2 x 1 (which is 24) 24 = 2^3 x 3 Therefore, 20 x 21 x 22 x 23 must also be divisible by 2^3 x 3, as well as all other factor combinations !

Question 29

What's important to know when you are asked what number divides into n^3 - n ?

Answer 29

That you can see that n^3 - n = n(n^2 - 1) = n(n+1)(n-1) Those are 3 consecutive terms, and must thus be divisible by 3!

Question 30

How would you tackle this?

Answer 30

Factorize the equation first! Spot how many consecutive terms there are!

Question 31

What's import when you are asked to know what number divides into n^2 - 1?

Answer 31

n^2 - 1 = (n-1)(n+1) That you SEE THIS, which, if n is odd, that this expression would be a product of TWO consecutive, EVEN integers

Question 32

The product of n consecutive even integers is divisible by __

Answer 32

2^n x n!