find the smallest natural number that we need to divide 1331 with to make it a perfect square
The correct answer and explanation is:
Let’s analyze the problem step by step:
Problem:
Find the smallest natural number by which we must divide 1331 to make the result a perfect square.
Step 1: Understand the problem
We want to find a natural number nn such that 1331n\frac{1331}{n} is a perfect square.
- A perfect square is a number that can be expressed as k2k^2 for some integer kk.
- We want to find the smallest natural number nn such that after dividing 1331 by nn, the result is a perfect square.
Step 2: Prime factorize 1331
Prime factorization helps us identify the factors clearly and find out what we need to divide by.
Check if 1331 is a perfect power or can be factored easily.
- 113=11×11×11=133111^3 = 11 \times 11 \times 11 = 1331
So, 1331=1131331 = 11^3
Step 3: Find what divides 1331 to make a perfect square
- A perfect square must have even powers of all prime factors.
- Currently, 1331=1131331 = 11^3, and the exponent 3 is odd.
- To make it a perfect square, the exponent of 11 in the quotient must be even.
Let’s suppose: 1331n=113÷n=11k\frac{1331}{n} = 11^{3} \div n = 11^k
where kk is an even integer (since the quotient must be a perfect square).
Since nn must be a natural number and a divisor of 1331, it can only be 11m11^m where m≤3m \leq 3.
Step 4: Try dividing by powers of 11 to make the exponent even
- If n=111n = 11^1, then: 113111=113−1=112\frac{11^3}{11^1} = 11^{3-1} = 11^2 11211^2 is 121121, which is a perfect square.
- If n=112n = 11^2, then: 113112=113−2=111=11\frac{11^3}{11^2} = 11^{3-2} = 11^1 = 11 11 is not a perfect square.
- If n=113n = 11^3, then: 113113=110=1\frac{11^3}{11^3} = 11^0 = 1 1 is a perfect square, but 1 is greater than dividing by 11111^1 (which is 11), so n=1331n=1331 is bigger than 11.
Step 5: Conclusion
The smallest natural number nn that divides 1331 to yield a perfect square is: 11\boxed{11}
Explanation Summary (300 words)
To determine the smallest natural number by which 1331 must be divided to become a perfect square, we begin with prime factorization. The number 1331 is not arbitrary; it is equal to 11311^3. A perfect square requires that all prime factors appear with an even exponent. Since 1331 equals 11311^3, it has an odd exponent (3) for the prime factor 11.
To make 1331n\frac{1331}{n} a perfect square, the division must adjust the exponent of 11 to an even number. Specifically, after dividing by nn, the power of 11 in the quotient must be even. Considering nn must be a divisor of 1331, it must be a power of 11: either 11111^1, 11211^2, or 11311^3.
If we divide 1331 by 111=1111^1 = 11, the quotient is 113−1=112=12111^{3-1} = 11^2 = 121, which is a perfect square. Dividing by 112=12111^2 = 121 yields 113−2=111=1111^{3-2} = 11^1 = 11, which is not a perfect square. Dividing by 113=133111^3 = 1331 yields 1, which is a perfect square but results in the largest divisor, not the smallest.
Therefore, the smallest divisor that turns 1331 into a perfect square when divided is 1111.
This approach of using prime factorization and ensuring even exponents for perfect squares is a standard method for solving such problems. It simplifies the process by focusing on the properties of exponents rather than testing arbitrary divisors.