Math Test 2
The largest number that will divide 398,436 and 542 leaving remainders 7,11 and 15 respectively is
Explanation: To find the largest number that will divide 398, 436, and 542 leaving remainders of 7, 11, and 15 respectively, we can subtract the remainders from the numbers and then find the greatest common divisor (GCD) of these differences.
Let's calculate:
1. For 398: 398−7=391398 - 7 = 391398−7=391
2. For 436: 436−11=425436 - 11 = 425436−11=425
3. For 542: 542−15=527542 - 15 = 527542−15=527
Now, we find the GCD of 391, 425, and 527.
The GCD of these numbers is 17.
So, the correct answer is:
(a) 17
There are 312, 260 and 156 students in class X, XI and XII respectively. Buses are to be hired to take these students to a picnic. Find the maximum number of students who can sit in a bus if each bus takes equal number of students
Explanation: To find the maximum number of students who can sit in a bus if each bus takes an equal number of students, we need to find the greatest common divisor (GCD) of the numbers of students in each class.
The GCD of 312, 260, and 156 can be found by taking the GCD of pairs of these numbers iteratively.
GCD(312,260)=GCD(260,52)=52\text{GCD}(312, 260) = \text{GCD}(260, 52) = 52GCD(312,260)=GCD(260,52)=52 GCD(52,156)=52\text{GCD}(52, 156) = 52GCD(52,156)=52
So, the maximum number of students who can sit in a bus is 52.
Therefore, the correct answer is:
(a) 52
There is a circular path around a sports field. Priya takes 18 minutes to drive one round of the field. Harish takes 12 minutes. Suppose they both start at the same point and at the same time and go in the same direction. After how many minutes will they
Explanation: To find out when Priya and Harish will meet, we need to find the least common multiple (LCM) of their time taken to complete one round of the field.
Priya takes 18 minutes to complete one round, and Harish takes 12 minutes.
The LCM of 18 and 12 is 36 minutes.
So, Priya and Harish will meet again after 36 minutes.
Therefore, the correct answer is:
(a) 36 minutes
Express 98 as a product of its primes
Explanation: To express 98 as a product of its prime factors, we need to find its prime factorization.
We can start by dividing 98 by the smallest prime number, which is 2:
98÷2=49
Now, we divide 49 by the next smallest prime number, which is 7:
49÷7=7
Now, since 7 is a prime number, we cannot further divide it.
So, the prime factorization of 98 is 2×72
Therefore, the correct answer is:
(c) 2×72
Three farmers have 490 kg, 588 kg and 882 kg of wheat respectively. Find the maximum capacity of a bag so that the wheat can be packed in exact number of bags.
Explanation: To find the maximum capacity of a bag so that the wheat can be packed in an exact number of bags, we need to find the greatest common divisor (GCD) of the amounts of wheat each farmer has.
The GCD of 490, 588, and 882 can be found by taking the GCD of pairs of these numbers iteratively.
GCD(490,588)=GCD(98,588)=GCD(98,490)=98\text{GCD}(490, 588) = \text{GCD}(98, 588) = \text{GCD}(98, 490) = 98GCD(490,588)=GCD(98,588)=GCD(98,490)=98 GCD(98,882)=98\text{GCD}(98, 882) = 98GCD(98,882)=98
So, the maximum capacity of a bag is 98 kg.
Therefore, the correct answer is:
(a) 98 kg
For some integer p, every even integer is of the form
Explanation: An even integer can be represented as 2n, where n is an integer.
So, for some integer p, every even integer is of the form 2p
Therefore, the correct answer is:
(b) 2p
For some integer p, every odd integer is of the form