Give the prime factorization of each of the following numbers :
945

Answer

To find the prime factorization of the number 945, we will divide it by the smallest prime numbers until we reach 1. Here’s the step-by-step solution:

### Step 1: Start with the number 945.
We will check if 945 is divisible by the smallest prime number, which is 2. Since 945 is an odd number, it is not divisible by 2.

**Hint:** Check if the number is even or odd to determine if it can be divided by 2.

### Step 2: Check divisibility by the next prime number, which is 3.
To check if 945 is divisible by 3, we can add the digits of 945 (9 + 4 + 5 = 18). Since 18 is divisible by 3, we can divide 945 by 3.

\[ 945 ÷ 3 = 315 \]

**Hint:** A number is divisible by 3 if the sum of its digits is divisible by 3.

### Step 3: Divide 315 by 3 again.
Now we will check if 315 is divisible by 3. Adding the digits of 315 (3 + 1 + 5 = 9), we see that 9 is divisible by 3.

\[ 315 ÷ 3 = 105 \]

**Hint:** Repeat the process of checking divisibility by summing the digits.

### Step 4: Divide 105 by 3 once more.
Next, we check if 105 is divisible by 3. Adding the digits of 105 (1 + 0 + 5 = 6), we find that 6 is divisible by 3.

\[ 105 ÷ 3 = 35 \]

**Hint:** Continue dividing by 3 as long as the sum of the digits is divisible by 3.

### Step 5: Now check 35 for divisibility by the next prime number, which is 5.
Since 35 ends in 5, it is divisible by 5.

\[ 35 ÷ 5 = 7 \]

**Hint:** A number is divisible by 5 if it ends in 0 or 5.

### Step 6: Finally, divide 7 by itself.
Since 7 is a prime number, we can divide it by 7.

\[ 7 ÷ 7 = 1 \]

**Hint:** Any prime number is only divisible by 1 and itself.

### Step 7: Collect all the prime factors.
Now we can write down all the prime factors we used:

- From 945: \( 3 \)
- From 315: \( 3 \)
- From 105: \( 3 \)
- From 35: \( 5 \)
- From 7: \( 7 \)

Thus, the prime factorization of 945 is:

\[ 945 = 3^3 \times 5^1 \times 7^1 \]

**Final Answer:** The prime factorization of 945 is \( 3^3 \times 5 \times 7 \).