A number is cube of 53. When 7 times of 57 is subtracted from the number, then the resultant number which is formed will be divisible by:
22
Finding the Divisor of a Number Calculation
Let's break down this quantitative aptitude problem step-by-step. We are given a number which is the cube of 53. From this number, we subtract 7 times 57. We need to find which of the given options divides the resultant number.
Step 1: Calculate the original number
The original number is the cube of 53. This means we need to calculate \(53^3\).
\[53^3 = 53 \times 53 \times 53\]First, let's calculate \(53 \times 53\):
\[53 \times 53 = 2809\]Now, let's multiply this result by 53 again:
\[2809 \times 53 = 148877\]So, the original number is 148877.
Step 2: Calculate the number to be subtracted
We need to subtract 7 times 57 from the original number. Let's calculate \(7 \times 57\).
\[7 \times 57 = 399\]The number to be subtracted is 399.
Step 3: Calculate the resultant number
The resultant number is obtained by subtracting 399 from 148877.
\[\text{Resultant Number} = 148877 - 399\] \[148877 - 399 = 148478\]The resultant number is 148478.
Step 4: Check divisibility of the resultant number by the given options
Now we need to check which of the options (18, 22, 15, 20) divides 148478 without leaving a remainder.
Divisibility by 18
A number is divisible by 18 if it is divisible by both 2 and 9. The number 148478 is an even number (ends in 8), so it is divisible by 2. To check for divisibility by 9, we sum the digits:
\[1 + 4 + 8 + 4 + 7 + 8 = 32\]Since the sum of the digits (32) is not divisible by 9, 148478 is not divisible by 9. Therefore, it is not divisible by 18.
Divisibility by 22
A number is divisible by 22 if it is divisible by both 2 and 11. The number 148478 is even, so it is divisible by 2. To check for divisibility by 11, we find the difference between the sum of the digits at odd places and the sum of the digits at even places (starting from the rightmost digit as place 1).
Digits at odd places (1st, 3rd, 5th from right): 8, 4, 4. Sum = \(8 + 4 + 4 = 16\).
Digits at even places (2nd, 4th, 6th from right): 7, 8, 1. Sum = \(7 + 8 + 1 = 16\).
Difference = Sum of digits at odd places - Sum of digits at even places = \(16 - 16 = 0\).
Since the difference is 0, 148478 is divisible by 11. Since it is divisible by both 2 and 11, it is divisible by 22.
Divisibility by 15
A number is divisible by 15 if it is divisible by both 3 and 5. The number 148478 ends in 8, so it is not divisible by 5. Therefore, it is not divisible by 15.
Divisibility by 20
A number is divisible by 20 if it is divisible by both 4 and 5. The number 148478 ends in 8, so it is not divisible by 5. Also, the number formed by the last two digits is 78, which is not divisible by 4. Therefore, it is not divisible by 20.
Based on the divisibility tests, the resultant number 148478 is divisible by 22.
| Option | Divisibility Test | Result |
|---|---|---|
| 18 | Divisible by 2 and 9? | No (not by 9) |
| 22 | Divisible by 2 and 11? | Yes (by both) |
| 15 | Divisible by 3 and 5? | No (not by 5) |
| 20 | Divisible by 4 and 5? | No (not by 5, not by 4) |
Thus, the resultant number is divisible by 22.
Revision Table: Number Properties and Divisibility
| Concept | Description | Example |
|---|---|---|
| Cube of a number | Multiplying a number by itself three times (\(n^3\)). | \(53^3 = 53 \times 53 \times 53\) |
| Divisibility by 2 | A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8). | 148478 is divisible by 2. |
| Divisibility by 9 | A number is divisible by 9 if the sum of its digits is divisible by 9. | Sum of digits of 148478 is 32, not divisible by 9. |
| Divisibility by 11 | A number is divisible by 11 if the alternating sum of its digits (starting from the right) is 0 or a multiple of 11. | For 148478: \(8-7+4-8+4-1=0\), divisible by 11. |
| Divisibility by Composite Numbers | To check divisibility by a composite number (like 18, 22, 15, 20), check if the number is divisible by its coprime factors. | 18 = 2 x 9 (check divisibility by 2 and 9) 22 = 2 x 11 (check divisibility by 2 and 11) 15 = 3 x 5 (check divisibility by 3 and 5) 20 = 4 x 5 (check divisibility by 4 and 5) |
Additional Information: Quantitative Aptitude Skills
Problems like this test your ability to perform basic arithmetic operations accurately and apply divisibility rules. Understanding divisibility rules can save a lot of time, especially in multiple-choice questions. Practicing cube calculations for smaller numbers is also helpful. For larger cubes or complex multiplications, being careful with calculations is crucial. Always double-check your arithmetic steps to avoid errors.
Important Questions from Cube and Cube Root
If 5 \(\sqrt{3}\) + \(\sqrt{75}\) = 17.32, then the value of 14 \(\sqrt{3}\) + \(\sqrt{108}\) is:
Find the value of (25)3 + (-29)3 + (4)3
The value of \(\frac{\sqrt[3]{-2744} \times \sqrt[3]{-216}}{\sqrt[3]{\frac{64}{729}}}\) is:
The sum of a positive number and its cube is 1740. What is the value of the number?
The largest four digit number which is a perfect cube is:
