Real Numbers Test

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Real Numbers Test - 4

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Weekly Quiz Competition

Question 1
1 / -0

If (a³ + k) is divisible by k, where 'a' is a positive integer and 'k' is a prime number, then which of the following is true?

A
k is a multiple of a.

B
k is a factor of a²but not a.

C
k is a factor of a.

D
k is a factor of a³and a but not a².

Solution

If (a³ + k) is divisible by 'k', then a³ is also divisible by 'k'.
Hence, 'k' is a factor of a³.

As 'k' is a prime number, 'a' needs to be the multiple of 'k'.
So, 'k' is a factor of a^b, where 'b' is any positive integer.
Question 2
1 / -0

A real number k/150, where 'k' is a three-digit number with digits 2, 7 and 3 in any order, is expressed as a decimal number. Which of the following is true for the number?

A
It is a non-terminating repeating decimal number.

B
It is a terminating decimal number.

C
It is a non-terminating, non-repeating decimal number.

D
Any of the above statement could be true.

Solution

Sum of the digits of the number (k) = 2 + 7 + 3 = 12
12 is divisible by 3, hence the number (k) is a multiple of 3.

Hence, 150 ÷ 3 = 50

As the denominator can be reduced to 50, the number is a terminating decimal number because 50 = 2 × 5²and if the prime factorisation of the denominator has only powers of 2 or 5 or both, then the number is a terminating decimal.
Question 3
1 / -0

There are three watches, out of which one loses 4 seconds every minute and one gains 4 seconds every minute. If all the watches are set at 8 a.m. and an alarm is activated, each watch beeps for a second when the time turns 1 minute in that watch. At what time will they beep together again?

A
9:52 a.m.

B
9:48 a.m.

C
9:58 a.m.

D
10:02 a.m.

Solution

In 1 minute, the first clock will cover 56 seconds.
The second one will cover 64 seconds and the third one will cover 60 seconds.
The time when they beep together will be the LCM of 56, 60 and 64, i.e. 6720.
The time they will beep together is 6720 seconds or 112 minutes (at 9:52 a.m.).
Question 4
1 / -0

A and B think about two numbers 'a' and 'b'. They say that the sum of their numbers is 270. If the HCF of the two numbers is 18, then what is the maximum number of such pairs which will satisfy these conditions?

A
10

B
9

C
6

D
4

Solution

Since 18 is the HCF of the two numbers, let the numbers be a = 18x and b = 18y, such that x and y are relatively prime.
a + b = 18x + 18y = 18(x + y) = 270

So, x + y = 15

We need to find all pairs of numbers such that they are relatively prime and their sum is 15.

(2, 13), (4, 11), (7, 8), and (14, 1) are the pairs satisfying the given conditions.
Question 5
1 / -0

Sham had a certain number of chocolates which he wanted to share among students of a school. At first he tried to distribute equally between two students, then among three, four, five and six students, but every time he was left with one chocolate. When he tried to distribute them among 7 students, he succeeded. What is the smallest number of chocolates he could have?

A
301

B
308

C
350

D
413

Solution

L.C.M of 2, 3, 4, 5, and 6 = 60
The number of chocolates would be of the form 60K + 1.

We have to try various values of K, so as to make it divisible by 7.
Starting from K = 1, we need to check unless we get a multiple of 7.

K = 5 makes it 301, which is the answer.
Question 6
1 / -0

Which of the following statements is correct?

A. If number X = pⁿ × q^m × r and number Y = p^n-1 × q^m+1 × r, then L.C.M(X, Y) = pⁿ × q^m+1 × r
B. Square root of any non-square number would be an irrational number.
C. H.C.F(1970, 218) = H.C.F(218, 8)
D. The product of primes is equal to a composite number.

A
A, C and D

B
B and D

C
Only C and D

D
Only A and D

Solution

A. Number X = pⁿ × q^m x r and number Y = p^n-1 × q^m+1 × r
Greatest power of p in both numbers is n.

Greatest power of q in both numbers is m + 1.
Greatest power of r in both numbers is 1.

Hence, L.C.M(X, Y) = pⁿ × q^m+1 × r

So, statement A is correct.

B. Square root of any non-square number can be a rational number or an irrational number, but the square root of a non-square integer would be an irrational number.

Hence, statement B is incorrect.

For example: Square root of 0.04 is 0.2, which is a rational number.

C. If a = bq + r, then H.C.F(a, b) = H.C.F(b, r), where r is less than b but greater than or equal to zero.
1970 = 218 × 9 + 8

Therefore, H.C.F(1970, 218) = H.C.F(218, 8)

Hence, statement C is correct.

D. The product of primes is equal to a composite number.

A composite number has more than 2 factors. When two prime numbers are
multiplied, the outcome would have 4 factors, which include the outcome number itself, 1 and the two prime numbers.

Hence, the product of the primes is equal to a composite number.
Question 7
1 / -0

Find the correct terms which will replace the letters M, N, O, and P in the given sentences.

A. Zero is not included in the set of …M… numbers.
B. The product of the …N… power of each prime factor involved in the numbers is called …O….
C. Square of an irrational number is …P… a rational number.

A
M = natural, N = greatest, O = L.C.M, P = not necessarily

B
M = real, N = smallest, O = H.C.F, P = definitely

C
M = natural, N = smallest, O = L.C.M, P = definitely not

D
M = real, N = greatest, O = H.C.F, P = definitely

Solution

A. Natural numbers start from 1 and go till infinity. Zero is not included in the set of natural number, but it is included in real numbers.

B. The product of the greatest power of each prime factor involved in the numbers is called L.C.M

C. Square of an irrational number is not necessarily a rational number. For example: √2 is an irrational number but its square, which is 2, is a rational number. But, square of π stays irrational.

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Real Numbers Test - 4

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Real Numbers Test - 4

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Question 1

k is a multiple of a.

k is a factor of a2 but not a.

k is a factor of a.

k is a factor of a3 and a but not a2.

Solution

Question 2

It is a non-terminating repeating decimal number.

It is a terminating decimal number.

It is a non-terminating, non-repeating decimal number.

Any of the above statement could be true.

Solution

Question 3

9:52 a.m.

9:48 a.m.

9:58 a.m.

10:02 a.m.

Solution

Question 4

10

9

6

4

Solution

Question 5

301

308

350

413

Solution

Question 6

A, C and D

B and D

Only C and D

Only A and D

Solution

Question 7

M = natural, N = greatest, O = L.C.M, P = not necessarily

M = real, N = smallest, O = H.C.F, P = definitely

M = natural, N = smallest, O = L.C.M, P = definitely not

M = real, N = greatest, O = H.C.F, P = definitely

Solution

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k is a factor of a²but not a.

k is a factor of a³and a but not a².