Understanding the Problem
The question asks us to find the maximum number of identical square pieces that can be cut from a rectangular wooden plank measuring 4.25 m in length and 3.4 m in width. The key requirements are that the square pieces should be of the largest possible size and that there should be no wastage of wood during the cutting process.
Converting Dimensions to a Common Unit
To perform calculations accurately, especially when finding the Greatest Common Divisor (GCD), it's helpful to convert the dimensions of the plank from meters to a smaller unit, such as centimeters.
- Length = 4.25 m = 4.25 \(\times\) 100 cm = 425 cm
- Width = 3.4 m = 3.4 \(\times\) 100 cm = 340 cm
Determining the Largest Square Size using GCD
For the square pieces to be the largest possible size and for there to be no wastage, the side length of each square must be the Greatest Common Divisor (GCD) of the plank's length and width. The GCD is the largest positive integer that divides both numbers without leaving a remainder.
We find the GCD of 425 and 340 using prime factorization:
- Prime factorization of 425: \(425 = 5 \times 85 = 5 \times 5 \times 17 = 5^2 \times 17\)
- Prime factorization of 340: \(340 = 10 \times 34 = (2 \times 5) \times (2 \times 17) = 2^2 \times 5 \times 17\)
To find the GCD, we identify the common prime factors and take the lowest power of each:
Common factors are 5 and 17. \(GCD(425, 340) = 5^1 \times 17^1 = 85\).
Thus, the side length of the largest square piece that can be cut is 85 cm.
Calculating the Total Number of Square Pieces
With the side length of the largest possible square determined (85 cm), we can now calculate how many squares fit along the length and the width of the plank.
- Number of square pieces along the length = \(\frac{\text{Plank Length}}{\text{Square Side Length}} = \frac{425 \text{ cm}}{85 \text{ cm}} = 5\)
- Number of square pieces along the width = \(\frac{\text{Plank Width}}{\text{Square Side Length}} = \frac{340 \text{ cm}}{85 \text{ cm}} = 4\)
The total number of square pieces is found by multiplying the number of pieces that fit along the length by the number of pieces that fit along the width.
Total Number of Square Pieces = (Number along length) \(\times\) (Number along width)
Total Number of Square Pieces = \(5 \times 4 = 20\).
Based on the standard mathematical approach for finding the largest square pieces with no wastage, the calculation yields 20 pieces. This result differs from the provided answer option of 45. The method detailed above correctly applies the concepts of GCD and dimensions to solve this type of problem.
Similar Questions
- If HCF of 768 and \(x^6y^2\) is \(32xy\) for natural numbers \(x \ge 2, y \ge 2\), then what is the value of \((x + y)\)?
- What is the HCF of \(2^{36}-1\) and \(2^{45}-1\)?
The HCF of \(x\) and \(y\) is \(H\). Consider the following statements in respect of the HCF of \(p=\frac{x^3 + y^3}{x^2-xy+y^2}\) and \(q=\frac{x^3-y^3}{x^2+xy+y^2}\) :
I. The HCF of \(p\) and \(q\) can be \(H\).
II. The HCF of \(p\) and \(q\) can be \(2H\).
Which of the statements given above is/are correct?
- What is the HCF of \(x^3 + y^3 + 3xy-1\) and \((x + y)^2 - 1\)?
- A number N is such that when divided by 4, 6, 7 or 9, it leaves 3 as remainder. What is the smallest 4-digit number that satisfies this property?
- What is the HCF of \(x^4 - 13x^2y^2 - 300y^4\), \(x^3 - 4x^2y - 4xy^2 - 5y^3\) and \(x^3 - 125y^3\)?
- Let \(N\) be a 5-digit number. When \(N\) is divided by 6, 12, 15, 24 it leaves respectively 2, 8, 11, 20 as remainders. What is the greatest value of \(N\)?
Let \(N\) be the least positive multiple of 11 that leaves a remainder of 5 when divided by 6, 12, 15, 18. Which one of he following is correct?
Let \(n\) be a natural number. The HCF of \(n, n + 10\) is 10. If the LCM is \(x\) (a 2-digit number), then how many values of \(x\) are possible?
Consider the following statements in respect of prime numbers p and q:
- I. Their LCM is always an odd number.
- II. Sum of their LCM and HCF is always an even number.
Which of the statements given above is/are correct?
Important Questions from LCM and HCF
The HCF and LCM of two numbers are 12 and 72, respectively. If the ratio of the two numbers is 2 ∶ 3, then the larger of the two numbers is:
Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case.
Joseph visits the club on every 5 th day, Harsh visits on every 24 th day, while Sumit visits on every 9 th day. If all three of them met at the club on a Sunday, then on which day will all three of them meet again?
What is the least number which when divided by 12,20 and 24 leaves in each case a remainder of 8?
Which of the following is a pair of co-primes?
