NUMBER SYSTEM TEST

1.The product of two numbers is 120 and the sum of their squares is 289. The sum of the number is

A. 20                                             B. 23

C. 27                                             D. 150

ANSWER & EXPLANATION

ANSWER:B
Explanation
Discuss the solution
Given product 2 numbers and sum of their squares,
apply, (a + b)2 = (a2 + b2 + 2ab)
(a + b)2 = 289 + 2 * 120 = 529 => a+b = 23.

2. Find the remainder when 3164 is divided by 162?.

A.82                                              B. 81

C. 83                                             D. 80

ANSWER & EXPLANATION

ANSWER:B
Solution:
For x=4, Remainder 3x81=0
And, Remainder 3x162=81

3. What is the unit's digit of the number 6256–4256

A. 0.                                                B.1

C. 4                                                 D. 7

ANSWER & EXPLANATION

ANSWER: A
Since the exponents are even, we can apply the property that,
If 'x' is even ax–bx is always divisible by (a+b).
6256-4256 will always be divisible by (6+4)=10.
Now any number multiplied by 10 gives the last digit as 'zero'.
Alternative:
The last digit of both the number are same as '6'
Thus after subtracting the unit's digit be '0'.

4. If 123x4 is divisible by 4, then the digit in place of x is :

A. 1                                                   B.3

C. 0                                                   D.7

ANSWER & EXPLANATION

ANSWER: C
12314 is not divisible by 4
12334 is not divisible by 4
12304 is divisible by 4.

5. What is the four digit number in which the first digit is 1/3 of the second, the third is the sum of the first and second, and the last is three times the second?

A. 1349                                              B.6286

C.2686                                               D. 1341

ANSWER & EXPLANATION

ANSWER: A
First digit is 1/3 second digit => The numbers can be 1 & 3, 2& 6, 3 & 9.
First + second = third => we can eliminate 3 & 9 since 3 + 9 = 12.
Last is 3 times the second =>
we can eliminate option 2 & 6 since 3 * 6 = 18.
Hence the number is 1349.

6.How many numbers from 10 to 50 are exactly divisible by 3.

A. 13                                                    B.12

C. 14                                                    D. 11

ANSWER & EXPLANATION

ANSWER: B
12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45,48.
13 Numbers.
10/3 = 3 and 50/3 = 16 ==> 16 - 3 = 13. Therefore 13 digits.

7 . . The sum of the digits of a two-digit number is 15 and the difference between the digits is 3. What is the two-digit number?

A. 69                                           
B. 78
C. 96                                             
D.Cannot be determined

ANSWER & EXPLANATION

ANSWER :D
Explanation:
Let the ten's digit be x and unit's digit be y.
Then, x + y = 15 and x - y = 3 or y - x = 3.
Solving x + y = 15 and x - y = 3, we get: x = 9, y = 6.
Solving x + y = 15 and y - x = 3, we get: x = 6, y = 9.
So, the number is either 96 or 69.
Hence, the number cannot be determined.

8.What is the sum of two consecutive even numbers, the difference of whose squares is 84?

A. 34                                                    B. 38

C. 42                                                    D. 46

ANSWER & EXPLANATION

ANSWER: C
Explanation:
Let the numbers be x and x + 2
Then, (x + 2)2 - x2 = 84
4x + 4 = 84
4x = 80
x = 20
The required sum = x + (x + 2) = 2x + 2 = 42

9.Find the greatest five digit number that is exactly divisible by 7, 10, 15, 21 and 28.

A. 99840                                                    B. 99900

C. 99960                                                    D. 99990

ANSWER & EXPLANATION

ANSWER: C
The number should be divisible by 10 (2, 5), 15 (3, 5), 21 (3, 7), and 28 (4, 7).
Hence, it is enough to check whether the number is divisibile by 3, 4, 5 and 7.
Test of divisibility by 3: Sum of the digits will be divisible by 3 if a number is divisible by 3.
Test of divisibility by 4: The righmost two digits viz., the units and tens digits will be divisible by 4 if a number is divisible by 4. For e.g, 1232 is divisible by 4 because 32 is divisible by 4.
Test of divisibility by 5: The units digit is either 5 or 0.
99960 is the only number which is divisible by 3, 4, 5 and 7.
Correct answer choice (3)

10. How many factors of 1080 are perfect squares?

A.4                                                     B.6

C. 8                                                    D. 5

ANSWER & EXPLANATION

ANSWER: A
DETAILED SOLUTION
1080 = 23 * 33 * 5. For any perfect square, all the powers of the primes
have to be even numbers. So, if the factor is of the form 2a * 3b * 5c.
The values 'a' can take are 0 and 2, b can take are 0 and 2, and c can take the value 0.
Totally there are 4 possibilities. 1, 4, 9, and 36.
Correct Answer: 4 Possibilities.

11. The sum of the first 100 natural numbers, 1 to 100 is divisible by

A. 2, 4 and 8                                      B. 2 and 4

C. 2                                                    D. 100

ANSWER & EXPLANATION

ANSWER: C
The sum of the first 100 natural numbers is 101
is an odd number and 50 is divisible by 2
Hence, 50*101 will be divisible by 2