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- x, y are integers and x
^{2}<36 , 3y^{2}<17. What is the number of unique values of the product of xy?

- 10
- 17
- 18
- 23
- 55

Answer: A or C

Solution: This is a tricky one.

x^{2}<36 and 3y^{2} <17

If you are considering only positive values for x and y you will end up with option A or C as your answer.

But x = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 (11 values)

And y = -2, -1, 0, 1, 2 (5 values)

If you multiply you will get 11 x 5 = 55 option E but the question says UNIQUE values

Go step by step.

Multiply y = -2 with all 11 values of x and you will get 11 unique products

Multiply y = -1 with all 11 values of x and you will get only 6 additional unique products

For all other values of y you will not get any more unique products

Total: 11+6 = 17

- If a is divisible by 12 and b is divisible by 8, then which of the following must be true? Select all that apply

- ab is a multiple of 48
- 64 is a factor of ab
- ab is divisible by 16
- 36 is a factor of ab

Answer: A and C

Solution:

a = 12x and b = 8y

ab = 96xy

A: irrespective of the value of xy ab is divisible by 96 and hence 48

B: ab is equal to 96xy which is 32 x 3 times xy so while it is definitely divisible by 32 it’s divisibility by 64 depends on the values of x and y

C: ab is equal to 96xy which is 16 x 6 times xy so it is definitely divisible by 16

D: ab is equal to 96xy which is 8 x 6 x2 times xy so it is divisible by 36 only if xy provides another 3.

- When 25! is divided by 5
^{n}there is no remainder What is the greatest integer value of n?

Answer: B

Solution: 24! = 1 x 2 x 3 x ….x 23 x 24

The multiples of 5 in the product are 5, 10, 15, 20 and 25

Each of them produce one 5 except 25 which provides 2 5s

So there are 6 5s, n = 6

- N is divisible by 18

Quantity A

The number of unique factors of N

The number of unique factors of N

Quantity B

5

- if Quantity A is greater
- if Quantity B is greater
- if the two quantities are equal
- if the relationship between the two quantities cannot be determined

Answer: A

Solution: Even though we don’t know the number N we know it’s divisible by 18. SO, N = 18k where k = 1,2,3…..

For the least possible value, N = 18

18 has 6 factors {1, 2, 3, 6, 9, 18}

For larger values N will have more factors so in all cases quantity A is greater.

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- 3a + 5
- 3a – 9
- 3a + 10
- 3a – 4
- 3a + 4

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