4. Find the greatest number that divides 204 and 64 without a remainder. 5. Use Euclid's division algorithm to find the HCF of 455 and 42. 6. A sweet seller has 420 Kaju burfis and 130 Badam burfis. She wants to stack them in such a way that each stack has the same number and same type, and they take up the least area of the tray. What is the number of burfis that can be placed in each stack for this purpose?
Answer (1)
To solve these questions, we'll use concepts from number theory, specifically focusing on finding the greatest common divisor (GCD) or highest common factor (HCF).
Finding the greatest number that divides 204 and 64 without a remainder:
We'll find the GCD of 204 and 64 using their prime factorizations.
Prime factorization of 204: 204 = 2 2 × 3 × 17
Prime factorization of 64: 64 = 2 6
The common factor with the smallest power is 2 2 = 4 .
Therefore, the greatest number that divides both 204 and 64 without a remainder is 4 .
Using Euclid's division algorithm to find the HCF of 455 and 42:
Euclid's division algorithm involves repeated application of division to find the greatest common divisor by expressing the remainder in terms of the previous divisor and dividend.
Here's the step-by-step process:
455 ÷ 42 = 10 (quotient is 10, remainder is 35) 42 ÷ 35 = 1 (quotient is 1, remainder is 7) 35 ÷ 7 = 5 (quotient is 5, remainder is 0)
The remainder is now zero, so the last non-zero remainder is the HCF.
Therefore, the HCF of 455 and 42 is 7 .
A sweet seller has 420 Kaju burfis and 130 Badam burfis. She wants to stack them such that each stack has the same number and type, minimizing tray area:
To find the number of burfis per stack, we need to find the HCF of 420 and 130.
Using the Euclidean algorithm:
420 ÷ 130 = 3 (quotient is 3, remainder is 30) 130 ÷ 30 = 4 (quotient is 4, remainder is 10) 30 ÷ 10 = 3 (quotient is 3, remainder is 0)
The remainder is zero, so the last non-zero remainder is the HCF.
Therefore, the number of burfis per stack is 10 . The sweet seller can create stacks containing either 10 Kaju burfis or 10 Badam burfis each, minimizing the tray area used.
