Question

1. choose all the numbers that could be common denominators for \\(\frac{1}{9}\\) and \\(\frac{1}{2}\\)\

\\(\square\\) 11\
\\(\square\\) 16\
\\(\square\\) 18\
\\(\square\\) 36\
\\(\square\\) 45

Explanation:

Step1: Find LCM of 9 and 2

To find a common denominator for \(\frac{1}{9}\) and \(\frac{1}{2}\), we first find the least common multiple (LCM) of 9 and 2. The prime factorization of 9 is \(3\times3\) and of 2 is \(2\). So, LCM\((9, 2)= 2\times3\times3 = 18\).

Step2: Identify multiples of LCM

Common denominators are multiples of the LCM. So, we check which of the given numbers are multiples of 18.

• For 11: \(11\div18\) is not an integer, so not a multiple.
• For 16: \(16\div18\) is not an integer, so not a multiple.
• For 18: \(18\div18 = 1\), so it is a multiple.
• For 36: \(36\div18 = 2\), so it is a multiple.
• For 45: \(45\div18 = 2.5\), not an integer, so not a multiple. Wait, wait, 9 and 2: LCM is 18, but also, common denominators can be any common multiples. Wait, 9 and 2 are coprime? No, 9 and 2 have GCD 1, LCM is 18. But also, 45: 9 divides 45 (45÷9 = 5) and 2 divides 45? No, 45÷2 = 22.5. Wait, I made a mistake. Wait, common denominator must be divisible by both 9 and 2. So a number \(n\) must satisfy \(n\) divisible by 9 and \(n\) divisible by 2. So check divisibility by 9 and 2.

Divisible by 2: even numbers. Divisible by 9: sum of digits divisible by 9.

• 11: odd, not divisible by 2.
• 16: even, sum of digits 1 + 6 = 7, not divisible by 9.
• 18: even, 1 + 8 = 9, divisible by 9. So divisible by both.
• 36: even, 3+6 = 9, divisible by 9. Divisible by both.
• 45: odd, not divisible by 2. Wait, 45 is odd, so 2 does not divide 45. So 45 is out. Wait, earlier mistake: LCM is 18, but common denominators are multiples of LCM, so 18, 36, 54, etc. Also, any number that is a multiple of both 9 and 2. So 18: 18 is multiple of 9 (18÷9=2) and multiple of 2 (18÷2=9). 36: 36÷9=4, 36÷2=18. What about 45? 45÷9=5, 45÷2=22.5, so not divisible by 2. So 45 is out. Wait, the original problem: the last number is 45? Wait, the options are 11,16,18,36,45? Wait, maybe I misread. Wait, the fractions are \(\frac{1}{9}\) and \(\frac{1}{2}\). So common denominator must be divisible by 9 and 2. So:
• 11: no.
• 16: no.
• 18: yes (18/9=2, 18/2=9)
• 36: yes (36/9=4, 36/2=18)
• 45: 45/9=5, 45/2=22.5: no, so 45 is out. Wait, but maybe the problem was \(\frac{1}{9}\) and \(\frac{1}{2}\), so denominators 9 and 2. So common denominators are numbers divisible by both 9 and 2. So 18, 36, 54, etc. So 18 and 36 are correct. Wait, but earlier when I thought 45: 45 is divisible by 9 but not by 2, so not a common denominator. So the correct numbers are 18 and 36.

Answer:

18, 36 (the checkboxes for 18 and 36 should be selected)