Mr. Gonzales showed the student part of the prime factorization of 90. One factor is missing. What number completes this prime factorization? 2 times 6 times ?
The Correct Answer and Explanation is:
To find the missing factor in the prime factorization of 90, let’s first break down 90 into its prime factors.
- We start by dividing 90 by 2, the smallest prime number: 90÷2=4590 \div 2 = 4590÷2=45 So, 90=2×4590 = 2 \times 4590=2×45.
- Now, we look at 45. It’s divisible by 3 (since the sum of the digits, 4 + 5 = 9, is divisible by 3): 45÷3=1545 \div 3 = 1545÷3=15 So, 45=3×1545 = 3 \times 1545=3×15. Thus, we have 90=2×3×1590 = 2 \times 3 \times 1590=2×3×15.
- Now, we factor 15 further. 15 is divisible by 3 (since 1 + 5 = 6, which is divisible by 3): 15÷3=515 \div 3 = 515÷3=5 So, 15=3×515 = 3 \times 515=3×5. Therefore, the full prime factorization of 90 is: 90=2×3×3×590 = 2 \times 3 \times 3 \times 590=2×3×3×5
- Looking at your given expression: 2×6×?2 \times 6 \times ?2×6×?, we can see that 6 is made up of the factors 2 and 3 (since 6=2×36 = 2 \times 36=2×3). This means the missing factor must be 3×5=153 \times 5 = 153×5=15.
Thus, the missing factor to complete the prime factorization is 15.
