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If (543)6 = (317)k, find k?
Answer: 8
Given, (543)6 = (317)k,
∴ 5 × 62 + 4 × 6 + 3 × 1 = 3 × k2 + 1 × k + 7 × 1
⇒ 207 = 3k2 + k + 7
⇒ 3k2 + k – 200 = 0
Solving this quadratic equation, we get,
k = -25/3 or 8
Since base cannot be -ve.
∴ k = 8.
Hence, 8.
1331 is a perfect cube in which of the following bases?
Answer: D
Let us write (1331)n is decimal system.
∴ (1331)n = n3 + 3n2 + 3n + 1 = (n + 1)3
Whatever be the value of n, (1331)n is always a perfect cube.
Hence, option (d).
(abcd)13 – (dcba)13 is always divisible by?
Answer: D
Given, (abcd)13 – (dcba)13
∴ a × 133 + b × 132 + c × 13 + d – (d × 133 + c × 132 + b × 13 + a)
= a(133 - 1) + b (132 - 13) – c(132 - 13) – d(133 - 1)
Now each of these terms is divisible by 12, hence the number is also divisible by 12.
Hence, option (d).
A three-digit non-zero number (abc)5 when converted to base 7 becomes (cba)7. What is b?
Answer: 0
Given, (abc)5 = (cba)7
∴ 25a + 5b + c = 49c + 7b + a
⇒ 24a – 48c = 2b
⇒ b = 12(a – 2c)
Since b has to be a single digits number (a – 2c) has to be 0.
∴ b = 12 × 0 = 0
Hence, 0.
(B)12 + (BB)12 + (BBB)12 + (BBBB)12 = (…………..)10
Answer: 22616
(B)12 = 11 × 1 = 12 – 1.
(BB)12 = 11 × 12 + 11 = 122 - 1.
(BBB)12 = 11 × 122 + 11 × 12 + 11 = 123 - 1.
… and so on
∴ (B)12 + (BB)12 + (BBB)12 + (BBBB)12 = (12 - 1) + (122 - 1) + (123 - 1) + (124 - 1).
= (12 + 122 + 123 + 124) – 4 = 22616.
Hence, 22616.
If n = (111)2 + (222)3 + (333)4 + (444)5 + … + (FFF)16. Find n?
Answer: C
(111)2 = 1 × 22 + 1 × 21 + 1 = 23– 1.
Similarly, (222)3 = 33– 1,
… and so on till
(FFF)16 = 163 – 1.
∴ (111)2 + (222)3 + (333)4 + (444)5 + … + (FFF)16 = (23- 1) + (33 - 1) + (43- 1) + … + (163- 1)
= (23 + 23 + … + 163) - 15
= (13 + 23 + 23 + … + 163) - 16
- 16
Hence, option (c).
The number of three-digit numbers in septanary (base 7) system is?
Answer: 294
In base 7 we can use 7 digits i.e., 0, 1, 2, 3, 4, 5 and 6.
∴ Number of 3-digit numbers = 6 × 7 × 7 = 294.
Hence, 294.
Product of 44 and 11 represented when represented in a certain base is 3414. The number 3111 of this system, when converted to the decimal system is?
Answer: A
Given, 44 × 11 = (3414)n
∴ 484 = (3n3 + 4n2 + n + 4)
⇒ 3n3 + 4n2 + n = 480
This is possible when n = 5.
∴ (3111)5 = 3 × 125 + 25 + 5 + 1 = 406.
Hence, option (a).
The cube root of (20311)5 = (…………..)10.
Answer: 11
(20311)5 = 2 × 625 + 3 × 25 + 1 × 5 + 1 = 1331.
We know 113 = 1331.
∴ Cube root of 1331 = 11.
Hence, 11.
Is the number (27316)9 divisible by 8?
Type 1 if it is divisible.
Type 2 if it is not divisible.
Type 3 if it cannot be determined.
Answer: 2
To check if a number in base n is divisible by (n – 1), add all the digits and check if the sum is divisible by (n – 1) or not.
If the sum is divisible by (n – 1), then the number will also be divisible by (n – 1).
Here, sum of the digits = 2 + 7 + 3 + 1 + 6 = 19.
Since 19 is not divisible by 8 hence, the number is not divisible by 8.
Hence, 2.
