Quantitative Aptitude > Number System
RELATIONSHIPS BETWEEN NUMBERS MCQs
91 is divisible by 7. So, it is not a prime number.
\(\left(112\times5^{4}\right) = 112\times\left(\frac{10}{2}\right)^{4}=\frac{112\times10^{4}}{2^{4}}=\frac{1120000}{16}=70000\)
Let 232 = x. Then, (232 + 1) = (x + 1).
Let (x + 1) be completely divisible by the natural number N. Then,
(296 + 1) = [(232)3 + 1] = (x3 + 1) = (x + 1)(x2 - x + 1), which is completely
divisible by N, since (x + 1) is divisible by N.
23) 1056 (45
92
---
136
115
---
21
---
Required number = (23 - 21)
= 2.
1397 x 1397 = (1397)2
= (1400 - 3)2
= (1400)2 + (3)2 - (2 x 1400 x 3)
= 1960000 + 9 - 8400
= 1960009 - 8400
= 1951609.
132 = 4 x 3 x 11
So, if the number divisible by all the three number 4, 3 and 11, then the number is divisible by 132 also.
264\(\rightarrow\) 11,3,4 (/)
396\(\rightarrow\) 11,3,4 (/)
462 \(\rightarrow\)11,3 (X)
792 \(\rightarrow\) 11,3,4 (/)
968\(\rightarrow\) 11,4 (X)
2178 \(\rightarrow\) 11,3 (X)
5184\(\rightarrow\) 3,4 (X)
6336 \(\rightarrow\) 11,3,4 (/)
Therefore the following numbers are divisible by 132 : 264, 396, 792 and 6336.
Required number of number = 4.
Divisibility Rules:
• A number is divisible by 2 if its last digit is 0, 2, 4, 6 or 8.
• A number is divisible by 3 if the sum of its digits is divisible by 3.
• A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
• A number is divisible by 5 if its last digit is 0 or 5.
• A number is divisible by 6 if it is divisible by 2 and 3.
• A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
• A number is divisible by 9 if the sum of its digits is divisible by 9.
• A number is divisible by 10 if its last digit is 0.
• A number is divisible by 11 if the difference between the sum of its digits in the odd places and the sum of its digits in the even places is either 0 or a multiple of 11.
• A number is divisible by 12 if it is divisible by 3 and 4.
• A number is divisible by 14 if it is divisible by 2 and 7.
• A number is divisible by 15 if it is divisible by 3 and 5.
• A number is divisible by 18 if it is divisible by 2, 3 and 6.
Divisibility by 132:
A number is divisible by 132 if it is divisible by 3 and 11 and 4.
Now, let us check whether given numbers are divisible by 132 or not:
• 264: The number 264 is divisible by 3 as the sum of its digits (2 + 6 + 4) = 12 is divisible by 3. It is also divisible by 4 as the last two digits (64) are divisible by 4. Also, it is divisible by 11 as 11 divides the difference between the sum of its digits in the odd places (2 + 4) = 6 and the sum of its digits in the even places (6) = 6. Therefore, 264 is divisible by 132.
• 396: The number 396 is divisible by 3 as the sum of its digits (3 + 9 + 6) = 18 is divisible by 3. It is also divisible by 4 as the last two digits (96) are divisible by 4. Also, it is divisible by 11 as 11 divides the difference between the sum of its digits in the odd places (3 + 6) = 9 and the sum of its digits in the even places (9) = 9. Therefore, 396 is divisible by 132.
• 462: The number 462 is not divisible by 3 as the sum of its digits (4 + 6 + 2) = 12 is not divisible by 3. Therefore, 462 is not divisible by 132.
• 792: The number 792 is divisible by 3 as the sum of its digits (7 + 9 + 2) = 18 is divisible by 3. It is also divisible by 4 as the last two digits (92) are divisible by 4. Also, it is divisible by 11 as 11 divides the difference between the sum of its digits in the odd places (7 + 2) = 9 and the sum of its digits in the even places (9) = 9. Therefore, 792 is divisible by 132.
• 968: The number 968 is divisible by 3 as the sum of its digits (9 + 6 + 8) = 23 is divisible by 3. It is also divisible by 4 as the last two digits (68) are divisible by 4. Also, it is divisible by 11 as 11 divides the difference between the sum of its digits in the odd places (9 + 8) = 17 and the sum of its digits in the even places (6) = 6. Therefore, 968 is divisible by 132.
• 2178: The number 2178 is not divisible by 3 as the sum of its digits (2 + 1 + 7 + 8) = 18 is not divisible by 3. Therefore, 2178 is not divisible by 132.
• 5184: The number 5184 is divisible by 3 as the sum of its digits (5 + 1 + 8 + 4) = 18 is divisible by 3. It is also divisible by 4 as the last two digits (84) are divisible by 4. Also, it is divisible by 11 as 11 divides the difference between the sum of its digits in the odd places (5 + 8) = 13 and the sum of its digits in the even places (1 + 4) = 5. Therefore, 5184 is divisible by 132.
• 6336: The number 6336 is divisible by 3 as the sum of its digits (6 + 3 + 3 + 6) = 18 is divisible by 3. It is also divisible by 4 as the last two digits (36) are divisible by 4. Also, it is divisible by 11 as 11 divides the difference between the sum of its digits in the odd places (6 + 6) = 12 and the sum of its digits in the even places (3 + 3) = 6. Therefore, 6336 is divisible by 132.
Hence, out of the given eight numbers, four numbers (264, 396, 792, 5184) are divisible by 132. Therefore, the correct answer is Option A. 4.
If you think the solution is wrong then please provide your own solution below in the comments section .
935421 x 625 = 935421 x 54 = 935421 x \(\left(\frac{10}{2}\right)^{4}\)
= \(=\frac{935421\times10^{4}}{2^{4}} = \frac{9354210000}{16}\)
= 584638125