Write 1260, 2535 and 15625 as a product of prime factors.
The correct answer and explanation is:
To express the numbers 1260, 2535, and 15625 as a product of prime factors, we need to perform prime factorization on each number. Prime factorization involves dividing the number by prime numbers (2, 3, 5, 7, 11, etc.) until we can no longer divide evenly by any prime number.
1. Prime Factorization of 1260:
Start by dividing 1260 by the smallest prime number, 2, since 1260 is even.
1260 ÷ 2 = 630
Now, divide 630 by 2 again:
630 ÷ 2 = 315
Next, divide 315 by the next smallest prime, 3:
315 ÷ 3 = 105
Divide 105 by 3 again:
105 ÷ 3 = 35
Now divide 35 by the next smallest prime, 5:
35 ÷ 5 = 7
Finally, 7 is already a prime number.
Thus, the prime factorization of 1260 is:
1260 = 2² × 3² × 5 × 7
2. Prime Factorization of 2535:
Start by dividing 2535 by 3 (since the sum of digits of 2535 is divisible by 3):
2535 ÷ 3 = 845
Now divide 845 by 5 (since it ends in 5):
845 ÷ 5 = 169
Finally, 169 is a perfect square, and its square root is 13, which is prime:
169 ÷ 13 = 13
Thus, the prime factorization of 2535 is:
2535 = 3 × 5 × 13²
3. Prime Factorization of 15625:
15625 is not divisible by 2, 3, or 5, so we check divisibility by higher primes. It turns out that 15625 is a power of 5:
15625 ÷ 5 = 3125
3125 ÷ 5 = 625
625 ÷ 5 = 125
125 ÷ 5 = 25
25 ÷ 5 = 5
5 ÷ 5 = 1
Thus, the prime factorization of 15625 is:
15625 = 5⁶
Summary:
- 1260 = 2² × 3² × 5 × 7
- 2535 = 3 × 5 × 13²
- 15625 = 5⁶
Prime factorization breaks each number down into the prime numbers that multiply to give the original number. This method helps in simplifying calculations, such as finding the greatest common divisor or the least common multiple.