Jawahar Navodaya Vidyalaya Selection Test (JNVST)
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LCM and HCF
Questions (20)
Question 1: Find the HCF of 45, 75 and 165.
Answer: 15
Explanation:
To find the HCF, we need to find the common factors of the numbers. The prime factors are: 45 = 3 x 3 x 5, 75 = 3 x 5 x 5, 165 = 3 x 5 x 11. The common prime factors are 3 and 5. Therefore, the HCF is 3 x 5 = 15.
Question 2: Find the smallest number divided by 42, 98 and 70.
Answer: 1470
Explanation:
To find the smallest number that can be divided by 42, 98, and 70, we need to find the LCM. The prime factors are: 42 = 2 x 3 x 7, 98 = 2 x 7 x 7, 70 = 2 x 5 x 7. The LCM is 2 x 3 x 5 x 7 x 7 = 1470.
Question 3: Greatest number, which is to be divided by 280 and 1245 leaves the remainder 4 and 3 respectively, is
Answer: 138
Explanation:
The number when divided by 280 leaves a remainder of 4, and when divided by 1245 leaves a remainder of 3. This means the number is 280k + 4 and 1245m + 3. The greatest such number is the greatest common divisor (GCD) of 280 and 1245 minus 1, which is 138.
Question 4: Three bells ring at intervals of 12 min, 15 min and 18 min respectively. They started ringing simultaneously at 9 : 00 am. What will be the next time when they all ring simultaneously?
Answer: 12 : 00 pm
Explanation:
To find when they all ring together again, calculate the least common multiple (LCM) of 12, 15, and 18. The LCM is 180 minutes, which is 3 hours. So, they will ring together again at 12:00 pm.
Question 5: The greatest number which will divide 1277 and 1368 leaving 3 as the remainder in each case is
Answer: 97
Explanation:
If a number leaves a remainder of 3 when dividing both 1277 and 1368, then it divides (1277 - 3) = 1274 and (1368 - 3) = 1365 exactly. We need to find the greatest common divisor (GCD) of 1274 and 1365. The GCD of 1274 and 1365 is 97. Therefore, the greatest number is 97.
Question 6: LCM of 114 and 95 is
Answer: 570
Explanation:
To find the LCM of two numbers, we can use the prime factorization method or the formula LCM(a, b) = (a * b) / GCD(a, b). The prime factorization of 114 is 2 x 3 x 19, and for 95, it is 5 x 19. The GCD of 114 and 95 is 19. Using the formula: LCM(114, 95) = (114 * 95) / 19 = 570.
Question 7: Find the LCM of 12, 18 and 24.
Answer: 72
Explanation:
To find the LCM (Least Common Multiple) of 12, 18, and 24, we list the multiples of each number: - Multiples of 12: 12, 24, 36, 48, 60, 72, 84, ... - Multiples of 18: 18, 36, 54, 72, 90, ... - Multiples of 24: 24, 48, 72, 96, ... The smallest common multiple in these lists is 72. Therefore, the LCM of 12, 18, and 24 is 72.
Question 8: What is the greatest number that divides both 16 and 20 exactly?
Answer: 4
Explanation:
To find the greatest number that divides both 16 and 20 exactly, we need to find the greatest common divisor (GCD). The divisors of 16 are 1, 2, 4, 8, 16 and the divisors of 20 are 1, 2, 4, 5, 10, 20. The greatest common divisor is 4.
Question 9: The LCM of 12, 24 and 30 is
Answer: 120
Explanation:
To find the LCM (Least Common Multiple) of 12, 24, and 30, we list the prime factors of each number: - 12 = 2^2 x 3 - 24 = 2^3 x 3 - 30 = 2 x 3 x 5 The LCM is found by taking the highest power of each prime number that appears in the factorizations: - The highest power of 2 is 2^3 (from 24) - The highest power of 3 is 3 (from all numbers) - The highest power of 5 is 5 (from 30) Therefore, LCM = 2^3 x 3 x 5 = 8 x 3 x 5 = 120.
Question 10: What will be the HCF of 48, 144 and 576?
Answer: 48
Explanation:
The HCF (Highest Common Factor) is the largest number that divides all the given numbers. First, list the factors of each number: - Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 - Factors of 144: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144 - Factors of 576: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, 288, 576 The common factors are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The highest of these is 48.
Question 11: The HCF of two numbers is 38 and their LCM is 98154. If one of the number is 1558. The other number is
Answer: 2394
Explanation:
The product of the HCF and LCM of two numbers equals the product of the numbers. So, 38 x 98154 = 1558 x other number. Solving for the other number: other number = (38 x 98154) / 1558 = 2394.
Question 12: The HCF of two numbers is 38 and their LCM is 98154. If one of the number is 1558. The other number is
Answer: 2394
Explanation:
The relationship between the HCF (Highest Common Factor), LCM (Least Common Multiple), and the two numbers is given by the formula: HCF * LCM = Number1 * Number2. Here, HCF = 38, LCM = 98154, and one number is 1558. Let's find the other number: 38 * 98154 = 1558 * Number2. Solving for Number2 gives: Number2 = (38 * 98154) / 1558 = 2394. Therefore, the other number is 2394.
Question 13: Two brand of chocolate is available in 10 and 12 packing. If we want to purchase same number of chocolate for each brand, then atleast what number of pack should brought for each brand?
Answer: 6 pack of 10, 5 pack of 12
Explanation:
To find the least number of packs needed to have the same number of chocolates, we need to find the least common multiple (LCM) of 10 and 12. The LCM of 10 and 12 is 60. To get 60 chocolates, you need 6 packs of 10 (since 6 x 10 = 60) and 5 packs of 12 (since 5 x 12 = 60). Therefore, the correct option is '6 pack of 10, 5 pack of 12'.
Question 14: The number of numbers which are multiples of both 3 and 5 in the first 100 natural numbers is
Answer: 6
Explanation:
To find numbers that are multiples of both 3 and 5, we need numbers that are multiples of 15 (since 15 is the least common multiple of 3 and 5). 1. List multiples of 15 up to 100: 15, 30, 45, 60, 75, 90. 2. Count them: There are 6 numbers. Therefore, the answer is 6.
Question 15: Which of the following numbers is divisible by 3, 4, 5 and 6?
Answer: 60
Explanation:
A number divisible by 3, 4, 5, and 6 must be divisible by their least common multiple (LCM). The LCM of 3, 4, 5, and 6 is 60. Therefore, 60 is the number that is divisible by all these numbers.
Question 16: LCM of 42, 70, 98 and 126 is
Answer: 8820
Explanation:
To find the LCM (Least Common Multiple) of 42, 70, 98, and 126, we first find the prime factorization of each number: - 42 = 2 x 3 x 7 - 70 = 2 x 5 x 7 - 98 = 2 x 7 x 7 - 126 = 2 x 3 x 3 x 7 Next, we take the highest power of each prime number that appears in any of the factorizations: - The highest power of 2 is 2^1 - The highest power of 3 is 3^2 - The highest power of 5 is 5^1 - The highest power of 7 is 7^2 Now, multiply these together to get the LCM: LCM = 2^1 x 3^2 x 5^1 x 7^2 = 2 x 9 x 5 x 49 = 8820 Therefore, the LCM of 42, 70, 98, and 126 is 8820.
Question 17: The HCF and LCM of two numbers are 4 and 48 respectively. If one of these numbers is 12, the other number is
Answer: 16
Explanation:
To find the other number, we use the relationship between HCF, LCM, and the numbers: HCF * LCM = Number1 * Number2. Here, HCF = 4, LCM = 48, and one number is 12. So, 4 * 48 = 12 * Number2. Simplifying, we get 192 = 12 * Number2. Dividing both sides by 12 gives Number2 = 16.
Question 18: The sum of HCF and LCM of 45, 60 and 75 is
Answer: 330
Explanation:
First, find the HCF (Highest Common Factor) of 45, 60, and 75. The factors are: 45 = 3 × 3 × 5 60 = 2 × 2 × 3 × 5 75 = 3 × 5 × 5 The common factors are 3 and 5, so HCF = 3 × 5 = 15. Next, find the LCM (Lowest Common Multiple). The LCM is the product of the highest power of all prime factors: LCM = 2^2 × 3^2 × 5^2 = 900. The sum of HCF and LCM is 15 + 900 = 915.
Question 19: A common multiple of both 9 and 7 is A. This number is in between 1200 and 1300. What is number A?
Answer: 1260
Explanation:
To find a common multiple of 9 and 7, we need to find the least common multiple (LCM) of these two numbers. The LCM of 9 and 7 is 63. Now, we need to find a multiple of 63 that lies between 1200 and 1300. Dividing 1200 by 63 gives approximately 19.05, and 1300 by 63 gives approximately 20.63. The integer between these is 20. Multiplying 63 by 20 gives 1260, which is the number we are looking for.
Question 20: The HCF of two numbers 14 and 28 is 14. Find the LCM.
Answer: 28
Explanation:
The formula to find the LCM using HCF is: LCM(a, b) = (a * b) / HCF(a, b). Here, a = 14 and b = 28, and HCF = 14. So, LCM = (14 * 28) / 14 = 28.