(a) 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 find the prime factorization of 132, we start by dividing 132 by the smallest prime number, 2, since 132 is even:132÷2=66132 \div 2 = 66132÷2=66
Now, divide 66 by 2 again:66÷2=3366 \div 2 = 3366÷2=33
Next, divide 33 by the smallest prime number greater than 2, which is 3:33÷3=1133 \div 3 = 1133÷3=11
Finally, 11 is a prime number, so 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 value of k.
For a number to be a perfect square, each prime factor in its prime factorization must appear to an even power. The prime factorization of 132 is 22×3×112^2 \times 3 \times 1122×3×11, where the powers of 3 and 11 are both 1 (odd). To make the exponents of 3 and 11 even, we need to multiply by one more 3 and one more 11.
Thus, the smallest value of kkk that makes 132k a perfect square is:k=3×11=33k = 3 \times 11 = 33k=3×11=33
So, the smallest integer kkk is 33.
(c) ppp is a number between 250 and 350. The highest common factor of ppp and 132 is 22. Find the smallest possible value of ppp.
First, recall the prime factorization of 132: 132=22×3×11132 = 2^2 \times 3 \times 11132=22×3×11. For the highest common factor (HCF) of ppp and 132 to be 22, ppp must contain the prime factors of 132 that result in an HCF of 22. The prime factorization of 22 is 2×112 \times 112×11, so ppp must have at least one factor of 2 and one factor of 11, but no factors of 3.
Therefore, ppp must be a multiple of 22, but not a multiple of 3. The smallest multiple of 22 between 250 and 350 is:22×12=26422 \times 12 = 26422×12=264
So, the smallest possible value of ppp is 264.
Summary:
- (a) 132=22×3×11132 = 2^2 \times 3 \times 11132=22×3×11
- (b) The smallest value of kkk is 33.
- (c) The smallest possible value of ppp is 264.
