1. To test the conjecture "If a number is divisible by 4, it is always an even number," we can take three examples:
Example 1: 12. This number is divisible by 4 since 12 divided by 4 is equal to 3. It is also an even number since it can be divided by 2 without leaving a remainder.
Example 2: 21. This number is not divisible by 4 since 21 divided by 4 results in a remainder of 1. Therefore, the conjecture is incorrect in this case.
Example 3: 8. This number is divisible by 4 since 8 divided by 4 is equal to 2. It is also an even number since it can be divided by 2 without leaving a remainder.
Based on these examples, we can see that the conjecture is incorrect, as not all numbers divisible by 4 are even.
2. To test the conjecture "All multiples of 5 have a 5 in the ones place," we can take three examples:
Example 1: 10. This number is a multiple of 5 since 10 divided by 5 is equal to 2. However, it does not have a 5 in the ones place as the ones digit is 0.
Example 2: 25. This number is a multiple of 5 since 25 divided by 5 is equal to 5. It does have a 5 in the ones place.
Example 3: 45. This number is a multiple of 5 since 45 divided by 5 is equal to 9. It does have a 5 in the ones place.
Based on these examples, we can see that the conjecture is correct, as all multiples of 5 do indeed have a 5 in the ones place.
3. To test the conjecture "If a number has a 9 in the ones place, it is always divisible by 3," we can take three examples:
Example 1: 29. This number has a 9 in the ones place, but it is not divisible by 3 since 29 divided by 3 results in a remainder of 2. Therefore, the conjecture is incorrect in this case.
Example 2: 39. This number has a 9 in the ones place and is divisible by 3 since 39 divided by 3 is equal to 13.
Example 3: 49. This number has a 9 in the ones place, but it is not divisible by 3 since 49 divided by 3 results in a remainder of 1.
Based on these examples, we can see that the conjecture is incorrect, as not all numbers with a 9 in the ones place are divisible by 3.
4. To test the conjecture "The least common denominator of two fractions is always greater than the denominators of the fractions," we need to work with specific fractions.
Example 1: Fractions with denominators 4 and 6. The least common denominator is 12, which is greater than both 4 and 6. Therefore, the conjecture is correct in this case.
Example 2: Fractions with denominators 3 and 9. The least common denominator is 9, which is equal to one of the denominators. Therefore, the conjecture is incorrect in this case.
Example 3: Fractions with denominators 2 and 8. The least common denominator is 8, which is equal to one of the denominators. Therefore, the conjecture is incorrect in this case.
Based on these examples, we can see that the conjecture is incorrect, as the least common denominator may or may not be greater than the denominators of the fractions.
5. Conjecture: The product of two odd numbers is always odd.
To test this conjecture, we can take three examples:
Example 1: 3 * 3 = 9. The product of two odd numbers is odd.
Example 2: 5 * 7 = 35. The product of two odd numbers is odd.
Example 3: 9 * 11 = 99. The product of two odd numbers is odd.
Based on these examples, we can conclude that the conjecture is correct, as the product of any two odd numbers is indeed odd.
6. Testing a conjecture is similar to finding a statement to be true or false because it involves gathering evidence or examples that support or contradict the conjecture. Both processes require examining specific cases to determine the validity of the statement or conjecture.
However, testing a conjecture is different in that it involves making an educated guess or hypothesis about a general relationship or pattern, whereas finding a statement to be true or false typically involves assessing the truth value of a specific statement or statement form. Additionally, testing a conjecture often requires examining multiple cases or examples to make a conclusion, whereas finding a statement to be true or false may involve logical reasoning or proof techniques.