Question

Jane writes down the prime decomposition of 224 as 2ᵃ x b. Calculate the values of a and b.

Answer 1

Jane's prime decomposition of 224 expressed as 2^a x b is 2 to the power of 5 times 7. Therefore, the values of a and b are 5 and 7, respectively.

Step-by-step explanation:

Jane writes down the prime decomposition of 224 as 2a x b. To calculate the values of a and b, we first need to factorize 224 into prime factors.

1. Start by dividing 224 by the smallest prime number, which is 2: 224 ÷ 2 = 112.
2. Continue to divide by 2 until you can no longer do so: 112 ÷ 2 = 56, then 56 ÷ 2 = 28, next 28 ÷ 2 = 14, and finally 14 ÷ 2 = 7.
3. Since 7 is a prime number and it does not divide evenly into any number smaller than itself other than 1, we have found our last prime factor.

So 224 can be written as 25 x 7, which identifies the values of a and b as 5 and 7, respectively.