Application of problem solving strategies
• Find the smallest number by which 1620 must be divided to get a perfect square.

Answer

To find the smallest number by which 1620 must be divided to get a perfect square, we can follow these steps:

### Step 1: Prime Factorization of 1620
First, we need to perform the prime factorization of 1620. We can do this by dividing the number by the smallest prime numbers until we reach 1.

- Divide 1620 by 2:
\( 1620 ÷ 2 = 810 \)
- Divide 810 by 2:
\( 810 ÷ 2 = 405 \)
- Divide 405 by 3:
\( 405 ÷ 3 = 135 \)
- Divide 135 by 3:
\( 135 ÷ 3 = 45 \)
- Divide 45 by 3:
\( 45 ÷ 3 = 15 \)
- Divide 15 by 3:
\( 15 ÷ 3 = 5 \)
- Finally, divide 5 by 5:
\( 5 ÷ 5 = 1 \)

So, the prime factorization of 1620 is:
\[ 1620 = 2^2 \times 3^4 \times 5^1 \]

### Step 2: Identify the Exponents
Next, we look at the exponents of the prime factors:
- The exponent of 2 is 2 (which is even).
- The exponent of 3 is 4 (which is even).
- The exponent of 5 is 1 (which is odd).

### Step 3: Make Exponents Even
For a number to be a perfect square, all the exponents in its prime factorization must be even. Here, the exponent of 5 is odd. To make it even, we need to divide by 5 once.

### Step 4: Conclusion
Thus, the smallest number by which 1620 must be divided to get a perfect square is **5**.

### Summary
The answer is: **5**

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