வெல்லுங்கள் CSAT 2025 - 4: கொள்குறி வினா - விடைகள் - Numbers -Fractions- LCM & HCF

Questions & Answers (Mutiple Choice)
கொள்குறி வினா- விடைகள்
Numbers -Fractions- LCM & HCF

1. How many numbers are there between 1 and 100 that are divisible by 8 or contain the digit 8?
a. 25 b. 26 c. 27 d. 28

2. Find the number of pairs such that the sum of two numbers is 234 and their HCF is 39.
a. 1 b. 2 c. 3 d. 4

3. If a number is divided by 154, the remainder is 82, what will be the remainder if it is divided by 154 and 55?
a. 2 b. 4 c. 5 d. 7

4. If 1 is added to the denominator, the fraction becomes 1/2 and if 1 is added to Numerator the fraction becomes 1, find the sum of Numerator and Denominator.
a. 8 b. 7 c. 6 d. 5

5. The ratio of two numbers is 2 : 3 and their LCM is 54. The sum of two numbers is
a. 5 b. 18 c. 27 d. 45

6. In a library, half of the books are
story-books, three-fourth of the remaining books are reference books and the rest of the books are encyclopaedia. The number of story-books in the library is
a. 600 b. 1200 c. 2000 d. 2400

7. Find the unit digit of (1243)^(1234).
a. 3 b. 9 c. 7 d. 1

8. What is the remainder when 7^(342) is divided by 342?
a. 1 b. 2 c. 3 d. 4

9. The maximum number of boys among which both 793 mangoes and 1342 oranges can be equally divided is
a. 41 b. 51 c. 61 d. 71

10. If x = (567.547)/(0.0028), then the value of (567547)/(28) in terms of x is
a. 10x b. 100x c. 1000x d. x/10

11. The floor of a hall is 8.75m long and 5.25m wide. It is to be paved by the least number of square marble tiles. The required number of tiles is
a. 12 b. 15 c. 8 d. 10

12. Sixteen times the cube root of a number is equal to the number itself. The number is
a. 16 b. 32 c. 64 d. 8

13. The least number , which when divided by20, 25, 35 and 40 leaves remainders 13, 18, 28 and 33 respectively is
a. 1407 b. 1393 c. 1400 d. 1405

14. If it is known that the unit digit of a
2-digit number exceeds the tenth digit by 2 and the product of twice the number and the sum of the digits is equal to 288, then the number is
a. 24 b. 42 c. 46 d. 53

15. The number of prime factors to the expression (2^5)×(6^10)×(7^17)×(11^27) is
a. 59 b. 63 c. 67 d. 69

Answers

1. c 2. a 3. c 4.d 5. d

6. a 7. b 8. a 9. c 10. d

11. b 12. c 13. b 14. a 15. d

Explanation of answers

1. There are 12 numbers which are divisible by 8.
The numbers containing the digit 8 are
8, 18, 28, 38, 48, 58, 68,78, 88, 98, 80, 81, 82, 83, 84, 85, 86, 87, 89. Totally 19 numbers are there from 1 to 100 containing the digit 8. Only four numbers 8, 48, 80 and 88 which are which are divisible by 8 and containing the digit 8.
Hence the required answer is
12 + 19 - 4 = 27.

2. Let the least ratio of two numbers be
x : y. Since HCF is 39, 39(x +y) = 234.
So x + y = 6. The possible pairs of (x, y) are (1, 5), (2, 4) and (3,3). Except (1,5) other pairs are not least ratios.
So, the only possibility is 1×39 and 5×39.
Hence the numbers are 39 and 195.

3. Since 154 = 11×7×2 and 55 = 11×5,
HCF (154, 55) = 11. It is enough to find the remainder when 82 is divided by 11.
Hence the required remainder is 5.

4. Let the number be x/y.
By problem, x/(y+1) = 1/2 -> 2x = y + 1
(x+1)/y = 1 -> x + 1 = y
Solving the last displayed equations,
x = 2 ; y = 3
Hence x + y = 2+ 3 = 5.

5. If two numbers are in the ratio x : y, then LCM/(xy) = HCF
So HCF = 54/(2×3) = 9
Product of two numbers = LCM × HCF
= 54×9 = 486.
The numbers are 9×2 and 9×3.
Sum of the two Numbers = 18 + 27 = 45.

6. Let the number of books be ' 1 '
The story-books = 1/2.
Reference books = (3/4)[1 - 1/2]
Encylopedia books = [1 - (3/4)] [1 - 1/2]
= 1/8
So 1/8 part of the total number of books is given by 150.
Hence the number of story-books
= 150 ×(1/2) ÷ (1/8) = 600

7. Here we have to take account of unit digit of the base number 1243.
It is enough to find the unit digit of
3^(1234).

The following table will be helpful for finding the unit digit of 2^n, 3^n, 7^n and 8^n.
Divide 'n' by 4, obtain the remainder. Possible remainders are 0, 1, 2 and 3.
Using the remainders
Remainders
0 1 2 3
Unit digit
2^n 6 2 4 8
3^n 1 3 9 7
7^n 1 7 9 3
8^n 6 8 4 2

0^n, 1^n, 5^n and 6^n end with same as 0, 1, 5 and 6 respectively.
To find the unit digit of 4^n and 9^n, we consider 'n' is odd or even.
Unit digit
n is odd even
4^n 4 6
9^n 9 1

To find unit digit of 3^(1234), divide 1234 and obtain the remainder. The remainder is 2. Hence the unit digit of
(1243)^(1234) is the unit digit of 3^2 = 9

8. We know 343 = 7^3 = 342 + 1
So 7^(342) = (7^3)^114 = (343)^114
= (342+1)^114
This expansion contains 115 terms. Except last term 1^(114) all other terms are multiples of 342.
Hence the remainder when 7^(342) is divided by 342 is 1.

9. Find HCF between 793 and 1342
793 = 13 ×61 ; 1342 = 2 × 11× 61
Clearly HCF is 61.

10. Since [(567.547)/(0.0028)] ×(1000/10000)
= 567547/28
Hence (567547)/(28) = x/10.

11. Find HCF of 8.75 and 5.25
8.75 = 1.75 × 5 whereas
5.25 = 1.75 × 3
Clearly HCF is 1.75.
So the size of square marble tile is 1.75m.
Hence the required number of
Square marbke tiles is 5 × 3= 15

12. Let the number be N
By problem
16 (N^(1/3)) = N -> (16^3) (N) = N^3
So N^2 = 16^3 = (4^3)^2
Hence N = 4^3 = 64

13. Divisors 20, 25, 35, 40
Remainders 13, 18, 28, 33
Difference 7, 7, 7, 7
If there is common difference between divisors and remainders, then the required least number is
LCM - Common Difference
Here LCM (20, 25, 35, 40) = 1400
The correct answer is 1400 - 7 = 1393.

14. Let the tenth digit be y
So unit digit must be y + 2
The value of two digit number is
(10y + y + 2)
By problem
2 (10y + y + 2)(y + y + 2) =288
gives y = 2
Hence the 2-digit number is 24
Short cut
By trial and error
288/2 = 144 = 12 × 12 = 6 × 24
So the answer must be 24.

15. (2^5)×(6^10)×(7^17)×(11^27)
= (2^5)×[(2×3)^(10)]×(7^17)×(11^27)
= (2^5)×(2^10)×(3^10)×(7^17)×(11^27)
= (2^15)×(3^10)×(7^17)×(11^27)
The number of prime factors is
15 + 10 + 17 + 27 = 69

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