Find the smallest positive integer to multiply by 896 and create a perfect squared number
The Correct Answer and Explanation is:
To find the smallest positive integer that, when multiplied by 896, results in a perfect square, we need to factorize 896 and analyze its prime factorization.
Step 1: Prime Factorization of 896
First, we find the prime factorization of 896 by dividing it by the smallest primes:896÷2=448896 \div 2 = 448896÷2=448448÷2=224448 \div 2 = 224448÷2=224224÷2=112224 \div 2 = 112224÷2=112112÷2=56112 \div 2 = 56112÷2=5656÷2=2856 \div 2 = 2856÷2=2828÷2=1428 \div 2 = 1428÷2=1414÷2=714 \div 2 = 714÷2=7
Now we have 896=27×7896 = 2^7 \times 7896=27×7.
Step 2: Conditions for a Perfect Square
For a number to be a perfect square, every prime factor in its prime factorization must have an even exponent. This means that in the factorization of 896, we have:
- 272^727, which has an odd exponent (7), and
- 717^171, which also has an odd exponent (1).
Step 3: Making the Exponents Even
To make the exponents of the primes even, we need to multiply by:
- One more factor of 222 to make the exponent of 2 even: 212^121, and
- One more factor of 777 to make the exponent of 7 even: 717^171.
Step 4: Finding the Multiplier
So, we multiply 896 by 2×7=142 \times 7 = 142×7=14. This gives:896×14=28×72896 \times 14 = 2^8 \times 7^2896×14=28×72
Now, the exponents of both prime factors are even, so the product 896×14896 \times 14896×14 is a perfect square.
Step 5: Conclusion
Thus, the smallest positive integer to multiply by 896 to make it a perfect square is 14.
