The Product of a Positive Integer x and 377,910 is Divisible by 3,300 GMAT Problem Solving

Question: The product of a positive integer x and 377,910 is divisible by 3,300, then the least value of x is:

1. 10
2. 11
3. 55
4. 110
5. 330

Correct Answer: D

Solution and Explanation:

Approach Solution 1:

Given in the problem that: (377910* x)/3,300 = integer

Factorizing the divisor we get,
3300= 2^2 * 3* 5^2 * 11

Now, checking the divisibility of 377,910 by 2^2, we get that it is not divisible by 2. Since 2^2 = 4(as the last 2 digits, 10 is not divisible by 4). The value of x has 2 as its factor. So 377,910 divided by 2, so to 377910*x divided by 2^2, must contain 2 as one of its factors.

So, 377910 divided by 3: 3+7+7+9+1+0=27, thus 377,910 IS divisible by 3.
377910 divided by 5^2: 377910 IS divisible by 5 and not divisible by 25. So, x must have 5 as a factor.
377910 divided by 11: (7+9+0) - (3+7+1) = 5, 377910 is not divisible by 11. So, x must have 11 as a factor.

Hence, the least value of x = 2*5*11 = 110.
The correct answer is D.

Approach Solution 2:

This solution approach is much easier and less time-consuming. This sum can be solved using the shortcut formula.

To answer this problem, we need to find out the factors:

It is given, 377,910/3,300
This implies 37,791/330 (cancel out zeros)
This implies not further divisible by 10. Therefore, x must have 10.
This implies 37,791/33 (divisible by 3? Yes)
This implies 12,597/11 (divisible by 11? No)
This implies x must have at least 10 and 11.
Hence the answer is 110.

The correct answer is D.

Approach Solution 3:

It is given that the product of a positive integer xx and 377,910 is divisible by 3,300, then the least value of x is:
To answer this: 3300 is obviously a multiple of 10 and 100.
Keep in mind that 377910 is only a multiple of 10.
Hence we need a 10.
Thereafter, 3300 is clearly a multiple of 11, but 33910 is not.
So we need an 11.

This leaves 110 and 330 as possible choices. Since we can see that both 377910 and 3300 are multiples of 3, 110 is the correct answer.
The correct answer is D.

“The product of a positive integer x and 377,910 is divisible by 3,300, then the least value of x” - is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This question has been taken from the book “GMAT Official Guide 2021”. To solve the GMAT Problem Solving questions, the candidates must have the basic concept of mathematics. The candidates can explore the GMAT Quant practice papers to practice varieties of questions that will enable them to improve their mathematical knowledge.

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