For all positive integers x and y, the expression xy is defined as

Question

﻿For all positive integers x and y, the expression xΘy is defined as the least multiple of y that is greater than or equal to x. For example, 2Θ3 = 3 and 3Θ2 = 4. For how many different positive integers k is 20Θk = 30?

A. One
B. Two
C. Three
D. Four
E. Five

gmatophobia · Accepted Answer

30 must be a multiple of k, hence k is a factor of 30

Factorization of 30

30 = 3 * 2 * 5

Positive factors of 30 = 1, 2, 3, 5, 6, 10, 15, 30

k = 1, least multiple = 20 
 k = 2, least multiple = 20 
 k = 3, least multiple = 21 
 k = 4, least multiple = 20 
 k = 5, least multiple = 20 
 k = 6, least multiple = 24 
 k = 10, least multiple = 20 
 k = 15, least multiple = 30 
 k = 30, least multiple = 30

Hence for two values least multiple = 30

Option B

Kinshook · Answer

Given: For all positive integers x and y, the expression xΘy is defined as the least multiple of y that is greater than or equal to x. For example, 2Θ3 = 3 and 3Θ2 = 4.

Asked: For how many different positive integers k is 20Θk = 30?

20Θk = 30

Since 30 is a multiple of k, k is a factor of 30.

Factor of 30=2*3*5 = {1,2,3,5,6,10,15,30}

If k = 1; 20Θk = 21
If k = 2; 20Θk = 22
If k = 3; 20Θk = 21
If k = 5; 20Θk = 25
If k = 6; 20Θk = 24
If k = 10; 20Θk = 21
If k = 15; 20Θk = 30; Correct
If k = 30; 20Θk = 30; Correct

IMO B

JeffTargetTestPrep · Answer

30 = (2)(3)(5)

First, we can list the positive integers that can have 30 as their multiple:

1, 2, 3, 5, (2)(3) = 6, (2)(5) = 10, (3)(5) = 15, (2)(3)(5) = 30

Next, we can check what is the least multiple of each of the above numbers that is greater than or equal to 20.

Actually, we don’t need to check 1, 2, 3, 5, 6, and 10 because each of these numbers definitely has at least one multiple that is less than 30 and greater than or equal to 20. Why? Because, from 20 to 29 there are 10 consecutive integers, and any 10 consecutive integers always include multiples of 1, 2, 3,…, and 10.

So, we have:

20 Θ 15 = 30
20 Θ 30 = 30

Answer: B

MartyMurray · Answer

We are provided the following information:

For all positive integers x and y, the expression xΘy is defined as the least multiple of y that is greater than or equal to x. For example, 2Θ3 = 3 and 3Θ2 = 4.

We have to answer the following question:

For how many different positive integers k is 20Θk = 30?

Answering this question may seem to be a big task, but our goal will be to make it as simple as possible.

The first thing we can do is see that only integers that have 30 as a multiple need to be considered.

We could start to list those integers, but we can save even more time by realizing that only integers whose smallest multiple greater than or equal to 20 is 30 work.

Using the information that 30 = 2 x 3 x 5, we can quickly work our way from the greatest integers that have 30 as a multiple to smaller ones to see which ones work.

2 x 3 x 5 = 30 works since 20Θ30 = 30.

3 x 5 = 15 works, since 20Θ15 = 30.

2 x 5 = 10 does not work, since 20Θ10 = 20.

Nothing smaller than 10 will work either because the difference between 20 and 30 is 10. So, any integer smaller than 10 that has 30 as a multiple will also have a multiple between 20 and 30.

So, only two integers, 30 and 15, work.

The correct answer is (B).

diya7 · Answer

Can someone please explain how least multiple is calculated for x and y in the sum, like 3 and 2 = 4 how ?

Bunuel · Answer

xΘy is defined as the least multiple of y that is greater than or equal to x.

3Θ2 is equal to the least multiple of 2 that is greater than or equal to 3. The least multiple of 2 that is greater than or equal to 3 is 4.

Hope it helps.

chetan2u · Answer

20#k=30 tells us that there is no multiple of k between 20, inclusive, and 30, exclusive. 
Surely k is greater than 10.
Next k is a factor of 30. => 1,2,3,5,6,10,15,30
Only possibilities are 15 and 30.

Thus , two is the answer. 
B

Nik091495 · Answer

can anyone explain why k=6 is not appropriate?

DanTheGMATMan · Answer

Bc 30 is not the least multiple of 6 greater than 20. 24 is

sahil770 · Answer

Understanding the Definition

The expression xΘy represents the smallest multiple of y that is greater than or equal to x.

Applying the Definition to the Problem

We are given that 20Θk = 30. This means:

30 is a multiple of k.
30 is the least multiple of k that is greater than or equal to 20.
Finding Possible Values of k

Since 30 is a multiple of k, k must be a factor (divisor) of 30. The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, and 30.

Now we need to check which of these factors satisfy the condition that 30 is the least multiple of k that is greater than or equal to 20.

k = 1: 20Θ1 = 20 (not 30)
k = 2: 20Θ2 = 20 (not 30)
k = 3: 20Θ3 = 21 (not 30)
k = 5: 20Θ5 = 20 (not 30)
k = 6: 20Θ6 = 24 (not 30)
k = 10: 20Θ10 = 20 (not 30)
k = 15: 20Θ15 = 30 (This works! The least multiple of 15 greater than or equal to 20 is 30)
k = 30: 20Θ30 = 30 (This works! The least multiple of 30 greater than or equal to 20 is 30)
If we go further, we see that if we want 30 to be a multiple of k, and be greater than or equal to 20, then k must be a factor of 30 that is greater than 20/2=10.
Since k must be a factor of 30, and greater than 10, the factors that work are 15 and 30.

Answer

There are two different positive integers k (15 and 30) for which 20Θk = 30.

Therefore, the correct answer is (B).

KarishmaB · Answer

Since 30 is a multiple of k, k must be a factor of 30. 
30 = 2*3*5
30 is also the lowest multiple of k greater than or equal to 20.
So k will be a large factor of 30. Let's start with the largest. 
k can be 30.
k can be 15 because least multiple of 15 greater than 20 is 30 only.
Can k be 10? No least multiple of 10 greater than or equal to 20 is 20, not 30.
k cannot be any smaller now because all smaller factors will have multiples less than 20.

Answer (B)

For all positive integers x and y, the expression xy is defined as

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For all positive integers x and y, the expression xy is defined as

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