) Express 132 as the product of its prime factors. (b) The number 132k is a perfect square. Find the smallest positive integer of k. (c) p is a number between 250 and 350. The highest common factor of p and 132 is 22. Find the smallest possible value of p.
The Correct Answer and Explanation is:
Express 132 as the product of its prime factors.
To express 132 as the product of its prime factors, we start by dividing 132 by the smallest prime numbers.
- 132 ÷ 2 = 66 (132 is divisible by 2, since it’s even)
- 66 ÷ 2 = 33 (66 is divisible by 2)
- 33 ÷ 3 = 11 (33 is divisible by 3, the sum of digits 3 + 3 = 6, which is divisible by 3)
- 11 is a prime number, so we stop here.
Thus, the prime factorization of 132 is: 132=22×3×11132 = 2^2 \times 3 \times 11132=22×3×11
(b) The number 132k is a perfect square. Find the smallest positive integer of k.
For 132k to be a perfect square, all the prime factors must appear with even exponents. From part (a), we know the prime factorization of 132 is: 132=22×3×11132 = 2^2 \times 3 \times 11132=22×3×11
The exponents of 3 and 11 are both 1 (odd), so we need to multiply 132 by the smallest kkk that will make all exponents even.
To make the exponent of 3 even, we need to multiply by another 3.
To make the exponent of 11 even, we need to multiply by another 11.
Thus, the smallest value of kkk is: k=3×11=33k = 3 \times 11 = 33k=3×11=33
Therefore, the smallest positive integer kkk is 33.
(c) p is a number between 250 and 350. The highest common factor of p and 132 is 22. Find the smallest possible value of p.
We are given that the highest common factor (HCF) of ppp and 132 is 22. From part (a), the prime factorization of 132 is: 132=22×3×11132 = 2^2 \times 3 \times 11132=22×3×11
Since the HCF is 22, ppp must include the prime factors of 22, which is: 22=2×1122 = 2 \times 1122=2×11
This means that ppp must be divisible by 2 and 11, but not by 3 (since the HCF does not include 3).
To find the smallest ppp between 250 and 350 that satisfies this condition, we start by checking multiples of 22 within that range.
The multiples of 22 between 250 and 350 are: 22×12=264and22×13=28622 \times 12 = 264 \quad \text{and} \quad 22 \times 13 = 28622×12=264and22×13=286
We check their divisibility by 3:
- 264÷3=88264 \div 3 = 88264÷3=88, so 264 is divisible by 3.
- 286÷3=95.33286 \div 3 = 95.33286÷3=95.33, so 286 is not divisible by 3.
Thus, the smallest possible value of ppp is 286.
Final Answers:
(a) The prime factorization of 132 is 22×3×112^2 \times 3 \times 1122×3×11.
(b) The smallest kkk for which 132k is a perfect square is k=33k = 33k=33.
(c) The smallest possible value of ppp is 286.
