Multiple Choice Questions
Number Theory and Mathematical Logic
Topic: Number Theory
Grade: 11
Question 1:
What is the remainder when 5^100 is divided by 7?
A) 0
B) 1
C) 2
D) 3
Answer: C) 2
Explanation: This question can be solved using the concept of modular arithmetic. We can notice that 5^2 ≡ 4 (mod 7), 5^4 ≡ 2 (mod 7), and 5^8 ≡ 4 (mod 7). Since 100 is divisible by 4, we can conclude that 5^100 ≡ 2 (mod 7). Therefore, the remainder when 5^100 is divided by 7 is 2.
Example:
Let\’s consider a simpler example. What is the remainder when 5^5 is divided by 7? Using the same logic as above, we can find that 5^5 ≡ 4 (mod 7). Therefore, the remainder is 4.
Example 2:
Now, let\’s consider a more complex example. What is the remainder when 5^17 is divided by 7? Since 17 is not divisible by 4, we need to calculate it step by step. 5^2 ≡ 4 (mod 7), 5^4 ≡ 2 (mod 7), 5^8 ≡ 4 (mod 7), 5^16 ≡ 2 (mod 7), and finally 5^17 ≡ 10 ≡ 3 (mod 7). Therefore, the remainder is 3.
Question 2:
How many positive integers less than 100 are divisible by either 2 or 3?
A) 32
B) 49
C) 50
D) 66
Answer: D) 66
Explanation: To find the number of positive integers less than 100 that are divisible by either 2 or 3, we need to find the union of the sets of numbers divisible by 2 and divisible by 3. There are 50 positive integers divisible by 2 (from 2 to 100 with a step of 2), and there are 33 positive integers divisible by 3 (from 3 to 99 with a step of 3). However, we need to subtract the numbers divisible by both 2 and 3 (multiples of 6) to avoid double counting. There are 16 multiples of 6 from 6 to 96 with a step of 6. Hence, the total number of positive integers less than 100 that are divisible by either 2 or 3 is 50 + 33 – 16 = 66.
Example:
Let\’s consider a simpler example. How many positive integers less than 10 are divisible by either 2 or 3? The numbers divisible by 2 are 2, 4, 6, 8, and the numbers divisible by 3 are 3 and 6. However, we need to subtract the number 6, which is divisible by both 2 and 3. Therefore, the total is 4.
Example 2:
Now, let\’s consider a more complex example. How many positive integers less than 1000 are divisible by either 4 or 5? The numbers divisible by 4 are from 4 to 996 with a step of 4, which gives us 249 numbers. The numbers divisible by 5 are from 5 to 995 with a step of 5, which gives us 199 numbers. However, we need to subtract the numbers divisible by both 4 and 5 (multiples of 20), which are from 20 to 980 with a step of 20, giving us 49 numbers. Therefore, the total is 249 + 199 – 49 = 399.