Let's solve the problem of finding the smallest number by which each given number must be multiplied to make it a perfect square. A perfect square has the property that all the prime factors in its prime factorization occur an even number of times.
To determine this smallest multiplier for any given number, we need to examine its prime factorization and see which primes occur an odd number of times. The product of these primes will be the smallest number by which we should multiply the original number to get a perfect square.
Let’s go through each of the provided numbers step by step:
### (a) 882
1. Perform prime factorization of 882:
- 882 = 2 × 3^2 × 7 × 7
2. Look for primes that occur an odd number of times:
- 2 (once)
- 7 (once)
3. The smallest number to multiply 882 to make it a perfect square:
- 2 × 7 = 14.
### (b) 432
1. Perform prime factorization of 432:
- 432 = 2^4 × 3^3
2. Look for primes that occur an odd number of times:
- 3 (occurs thrice)
3. The smallest number to multiply 432 to make it a perfect square:
- 3.
### (e) 3,698
1. Perform prime factorization of 3,698:
- 3,698 = 2 × 1849 = 2 × 43^2
2. Look for primes that occur an odd number of times:
- 2 (occurs once)
3. The smallest number to multiply 3,698 to make it a perfect square:
- 2.
### (c) 1,331
1. Perform prime factorization of 1,331:
- 1,331 = 11 × 11 × 11 = 11^3
2. Look for primes that occur an odd number of times:
- 11 (occurs thrice)
3. The smallest number to multiply 1,331 to make it a perfect square:
- 11.
### (f) 700
1. Perform prime factorization of 700:
- 700 = 2^2 × 5^2 × 7
2. Look for primes that occur an odd number of times:
- 7 (occurs once)
3. The smallest number to multiply 700 to make it a perfect square:
- 7.
### (d) 845
1. Perform prime factorization of 845:
- 845 = 5 × 169 = 5 × 13^2
2. Look for primes that occur an odd number of times:
- 5 (occurs once)
3. The smallest number to multiply 845 to make it a perfect square:
- 5.
### Summary
The smallest numbers by which the given numbers must be multiplied to make them perfect squares are:
- (a) 882 → 2
- (b) 432 → 3
- (e) 3,698 → 2
- (c) 1,331 → 11
- (f) 700 → 7
- (d) 845 → 5
