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Question: Rashi is going to plant 54 oak trees and 27 pine trees. Rashi would like to plant the trees in all rows that have the same number of trees and are made up of only one type of tree. What is the greatest number of trees Rashi can have in each row? What is the total number of rows she needs to plant? [4 MARKS]
Options:

: Concept: 1 Mark
Steps: 2 Marks
Answer: 1 Mark

We need to understand that if Rashi has to plant the same number of trees of the same type in a row, then the number of trees in each row should be a factor of 54 and 27.
Now, the greatest common factor will give the greatest number of trees Rashi can plant in each row.
The prime factorization of the numbers is given below. 54= 2 × 3 × 3 × 3 27= 3 × 3 × 3 So, the HCF will be 3 × 3 × 3 = 27 Therefore 27 trees can be planted in each row.
Total number of rows which will be planted = 54 ÷ 27 + 27÷27 = 3

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More Questions on This Topic :

Question 1. Define odd numbers. [1 MARK]

:
An odd number is anintegerwhich is not amultipleoftwo.
For example 1,3,5,7...
Question 2. Determine the least number which when divided by 3, 4 and 5 leaves a remainder of 2 in each case. [3 MARKS]

: Concept: 1 Mark
Steps: 1 Mark
Answer: 1 Mark

The LCM of 3, 4 and 5 = 3 × 4 × 5 = 60
Therefore, the required number is 2 more than 60.
Hence, the required least number = 60 + 2 = 62.
Question 3. Boxes that are 12 inches tall are being stacked next to boxes that are 18 inches tall. What is the shortest height at which the two stacks will be of same height? [4 MARKS]

: Concept: 1 Mark
Steps: 2 Marks
Answer: 1 Mark

For the shortest height, we need to find the LCM of 12 and 18 The prime factorization of 12 and 18 are: 12 = 2 × 2 × 3; 18 = 2 × 3 × 3 In these prime factorizations, the maximum number of times the prime factor 2 occurs is two; this happens for 12.
Similarly, the maximum number of times the factor 3 occurs is two; this happens for 18.
The LCM of the two numbers are the product of the prime factors counted the maximum number of times they occur in any of the numbers.
Thus, in this case LCM = 2 × 2 × 3 × 3 = 36.
Hence, the shortest height for which the two stacks will have the same height is 36 inches.
Question 4. If a number 3A98 is divisible by 11, find the value of A. [4 MARKS]

: Concept: 1 Mark
Steps: 2 Marks
Answer: 1 Mark

To check the divisibility of a number by 11, the rule is to find the difference between the sum of the digits at odd places (from the right) and the sum of the digits at even places (from the right) of the number. If the difference is either 0 or divisible by 11, then the number is divisible by 11.
3+ 9 = 12, A + 8 A + 8 - 12 = 0
A - 4 = 0 or A -4 = 11
A = 4 or A= 15, A = 15 is not possible. So, the number is 3498.
Question 5. State the divisibility rule for 4. Using divisibility test, determine which of the following numbers are divisible by 4?

(a) 4096             (b) 21084            

[2 MARKS]

: Each option: 1 Mark

A number is divisible by 4 if the last two digits of the whole numberare divisible 4.
(a) Since number formed by tens and units digit is 96, which is divisible by 4. Hence, 4096 is divisible by 4.
(b) Since number formed by tens and units digit is 84, which is divisible by 4. Hence, 21084 is divisible by 4.
Question 6. Akashi was playing with the numbers. She found that a number is divisible by both 7 and 12.

Which is the least number of which both of these will be divisible? [3 MARKS]

: Steps: 2 Marks
Solution: 1 Mark

The lowest number which will be divisible by both of these numbers is the LCM of these numbers.
To find the LCM of the numbers first we will find out its prime factors
The factors of 7: 1, 7
The factors of 12: 1, 2, 3, 4, 6, 12 Since the only common factor is 1, the given two numbers are co-prime.
The LCM of these numbers is their product.
The required number is 7×12 = 84.

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