Find the largest number which divides 438 and 606, leaving remainder 6 in each case.

Prime Factors

Prime factors are the prime numbers that divide a given number evenly, without leaving any remainder. Every positive integer greater than 1 can be expressed as a product of prime factors.

To determine the prime factors of a number, you divide the number by prime numbers starting from 2, continuing with 3, 5, 7, and so on, until the quotient becomes 1.

For Example: Find prime factors of 24.

Step 1: We start dividing the number by the smallest prime number, which is 2:

24 ÷ 2 = 12

Step 2: We continue dividing the quotient by the next prime number, which is 2 again:

12 ÷ 2 = 6

Step 3: Dividing by 2 once more:

6 ÷ 2 = 3

Step 4: The quotient 3 is a prime number itself.

Therefore, the prime factors of 24 are 2, 2, 2, and 3.

Expressing the number 24 as a product of its prime factors, we have 2 × 2 × 2 × 3.

Prime factors: 24 = 2^3 × 3.

Solution

To find the largest number that divides both 438 and 606, leaving a remainder of 6, we can subtract the remainder from both numbers and find the highest common factor (HCF) of the resulting numbers.

Step 1: Subtract the remainder (6) from both numbers:

432 = 438 – 6

600 = 606 – 6

Step 2: Find the HCF of 432 and 600.

Now, let’s find the HCF using the prime factorization method:

Prime factorization of 432: 2^4 * 3^3

Prime factorization of 600: 2^3 * 3 * 5^2

To find the HCF, we take the smallest power of common prime factors:

HCF = 2^3 * 3 = 8 * 3 = 24

Hence, the largest number that divides 438 and 606, leaving a remainder of 6, is 24.

Summary

Find the largest number which divides 438 and 606, leaving remainder 6 in each case.

The largest number which divides 438 and 606, leaving remainder 6 in each case is 24.

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