7TH NUMBERS TALENT - ODD, EVEN, PRIME AND COMPOSITE-WORD PROBLEMS - 3

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📝 Even, Odd, Prime & Composite Numbers Olympiad Quiz

1. [Even & Odd Numbers] The sum of four consecutive odd numbers is always divisible by which of the following fixed numbers?

(a) 4 (b) 6 (c) 8 (d) 16

Correct Answer: (c) 8

Let four consecutive odd numbers be (2k-3), (2k-1), (2k+1), and (2k+3).
Their sum is (2k-3) + (2k-1) + (2k+1) + (2k+3) = 8k.
Since the sum is 8k, it is always divisible by 8.

2. [Prime Numbers] If p and p + 2 are both prime numbers greater than 3, then their sum (p + p + 2) is always divisible by:

(a) 3 only (b) 6 only (c) Both 6 and 12 (d) 9

Correct Answer: (c) Both 6 and 12

For twin primes greater than 3 (e.g., 5 and 7, 11 and 13), the number between them is a multiple of 6. Let the middle number be 6k, so p = 6k - 1 and p + 2 = 6k + 1. Their sum is 12k, which is divisible by both 6 and 12.

3. [Composite Numbers] What is the smallest composite number that can be expressed as the sum of two distinct prime numbers in more than two different ways?

(a) 24 (b) 30 (c) 36 (d) 42

Correct Answer: (b) 30

Let's check 30: 7 + 23 = 30, 11 + 19 = 30, and 13 + 17 = 30 (three different ways using distinct prime pairs). Thus 30 is the smallest such composite number.

4. [Even & Odd Numbers] If x is an odd integer and y is an even integer, which of the following expressions results in an odd integer?

(a) 3x + 2y (b) 2x + 4y (c) x^2 + y^2 + 1 (d) x * y + x

Correct Answer: (a) 3x + 2y

3x is odd * odd = odd. 2y is even. Odd + even = odd.
Let's test with numbers: if x = 3, y = 2, then 3(3) + 2(2) = 9 + 4 = 13 (odd).

5. [Prime & Composite Numbers] How many prime numbers exist between 1 and 100 that contain the digit 3?

(a) 8 (b) 9 (c) 10 (d) 11

Correct Answer: (b) 9

Primes between 1 and 100 containing the digit 3 are: 3, 13, 23, 31, 37, 43, 53, 73, and 83. Counting them gives a total of 9 primes.

6. [Composite Numbers] Which of the following statements is universally true for any two composite numbers A and B?

(a) Their sum must be a composite number. (b) Their product must be a composite number. (c) Their difference must be an even number. (d) Their HCF must be greater than 1.

Correct Answer: (b) Their product must be a composite number.

Since both A and B have factors other than 1 and themselves, their product A * B will also have multiple factors, making it strictly a composite number.

7. [Even & Odd Numbers] If the product of five consecutive integers is negative, what is the maximum possible number of even integers among them?

(a) 1 (b) 2 (c) 3 (d) 4

Correct Answer: (c) 3

In any sequence of 5 consecutive integers, either 2 or 3 of them must be even. To maintain a negative product, we can have 3 even numbers with proper negative signs.

8. [Prime Numbers] Let x be a prime number such that x^2 + 1 is also a prime number. What can we conclude about x?

(a) x must be an odd prime greater than 3. (b) x must be 2. (c) x can be any prime number. (d) No such prime number exists.

Correct Answer: (b) x must be 2.

If x = 2, then x^2 + 1 = 2^2 + 1 = 5 (which is prime). If x is any odd prime, x^2 is odd, making x^2 + 1 even and greater than 2, which makes it composite. Thus, x must be 2.

9. [Composite Numbers] Which of the following numbers is the smallest composite number that has exactly 4 factors?

(a) 4 (b) 6 (c) 8 (d) 9

Correct Answer: (b) 6

Factors of 4: 1, 2, 4 (3 factors).
Factors of 6: 1, 2, 3, 6 (4 factors).
Factors of 8: 1, 2, 4, 8 (4 factors).
Since 6 is smaller than 8, 6 is the smallest composite number with exactly 4 factors.

10. [Even & Odd Numbers] If a, b, and c are three odd prime numbers, what is the nature of their sum (a + b + c)?

(a) Always an even number (b) Always an odd number (c) Always a prime number (d) Always a composite number

Correct Answer: (b) Always an odd number

All prime numbers greater than 2 are odd. The sum of three odd numbers is always odd (odd + odd = even, and even + odd = odd). Thus, a + b + c is always an odd number.