# Number system practice test

1.

If  3+4=21, 5+2=35, 4+2=24 what is the value of 5+3?

Explanation:

3+4=7*3 = 21

5+2=7*5 = 35

4+2=6*4 = 24

5+3=?

5+3=8*5=40

2.

Normally, numbers are written with a base of 10. Binary numbers have a base of 2. If 19257 = 3 x 7 x 7 x 131, in what base can it be written as 110100?

Explanation:

Let the base be x.

110100 is interpreted as x2 + x4 + x5

19257 = x2(1 + x2 + x3)

As 131 is a prime no. and 3 is also one, x = 7.

Therefore, 3 x 131 = 393 = 1 + x2 + x3, as 392 = x2 + x3 = 49 + 343 = 72 + 73.

3.

Six bells commence tolling together and toll at intervals of 2, 4, 6, 8 10 and 12 seconds respectively. In 30 minutes, how many times do they toll together?

Explanation:

L.C.M. of 2, 4, 6, 8, 10, and 12 is 120.
So, the bells will toll together after every 120 seconds (2 minutes).
In 30 minutes, they will toll together 30/2 + 1 = 16 times.

4.

Of the three numbers, second is twice the first and is also thrice the third, if the average of the three numbers is 44. Find the largest number?

Explanation:

Let the third number be x.
Then second number = 3x.
First number=3x/2.
Therefore x+3x + (3x/2) = (44*3) or x=24
So largest number = 2nd number = 3x = 72.

5.

Three times the first of three consecutive odd integers is 3 more than twice the third. The third integer is:

Explanation:

Let the three integers be x, x + 2 and x + 4. Then, 3x = 2(x + 4) + 3 x = 11. Third integer = x + 4 = 15.

6.

When 1100010 is divided by 0101, what will be the decimal remainder ?

7.

If a – b = 3 and a2 + b2 = 29, find the value of ab.

Explanation:

2ab = (a2 + b2) – (a – b)2 = 29 – 9 = 20

ab = 10.
8.

Find four consecutive even integers so that the sum of the first two added to twice the sum of the last two is equal to 742.

Explanation:

Let x, x + 2, x + 4 and x + 6 be the four integers. The sum of the first two
X + (x + 2)
Twice the sum of the last two is written as
2 ((x + 4) + (x + 6)) = 4 x + 20
Sum of the first two added to twice the sum of the last two is equal to 742 is written as
X + (x + 2) + 4 x + 20 = 742
Solve for x and find all four numbers
X = 120, x + 2 = 122, x + 4 = 124, x + 6 = 126
As an excise, check that the sum of the first two added to twice the sum of the last two is equal to 742.

9.

Find the base k of the number system, if (543)6=(317)k.

Explanation:

(543)6=5*62+4*6+3*60=(207)10
i.e., (207)10=(317)k
207=3k2+k+7=>3k2+k-200=0
=>3k2-24k+25k-200=0
=>(3k+25)(k-8)=0
=>k=8
Therefore the base is 8

10.

The square of a two digit number is divided by half the number. After 36 is added to the quotient, this sum is then divided by 2. The digits of the resulting number are the same as those in the original number, but they are in reverse order. The ten’s place of the original number is equal to twice the difference between its digits. What is the number?

Explanation:

Let the number be 10A + B= C
now, (C^2/(C/2) + 36)/2= 10B + A
or, (2C+36)/2 = 10B + A
or C+18=10B+A
or, 10A + B + 18 = 10 B + A
or, 18=9(B-A), giving B-A=2
As per the question, A=twice the difference between A and B
Hence A=2*2=4
B=2+4=6

The original number is 40+6=46.