How many positive integers less than 1000 (1) are divisible by 7? (2) are divisible by 7 but not by 11? (3) are divisible by both 7 and 11? (4) are divisible by either 7 or 11?

Q: If n is a prime number greater than 3, what is the remainder when n² is divided by 12?

A: The objective of this question is to find the remainder when the square of a prime number greater…

Q: Why k(k+1) is an even number?

A: We have to tell just  that why k(k+1) is an even number.Here we considering that you understood  the…

Q: Think about the three-digit numbers that can be formed from the digits 1, 2, 3, 4 and 5 without…

A: Solution

Q: 23. If 6p-4 is divisible by 7, where p is an integer, which of the following is also divisible by 7.…

A: Option a is correct answer

Q: 21 divides 4(n+1) + 5(2n-1) whenever n is a positive integer.

A: Steps and explanations are as follows:In case of any doubt, please let me know. Thank you.

Q: The number 42 has the prime factorization 2. 3. 7. Thus, 42 can be written in four ways as a product…

A: An integer 'n' is called a factor or a divisor of another integer 'm' if 'n' divides 'm' exactly…

Q: If x is divisible by 5 or y is divisible by 5, then xy is divisible by 5.

A: Let x is divisible by 5. Then x=5u for some natural number u.

Q: What are other similar expression patterns based on n(n+1)/2

A: Solution:-

Q: How many 6-digit numbers have three even and three odd digits? (The first digit cannot be zero.)

A: It is given that a 6-digit number is to be formed which will have three even and three odd digits.…

Q: A) (1 + √₂)² B) (1 -√2)² c)(1+√2)(1+√-2) D)(1+√2)(1-√√2) E)√2² + (√2)² Which of the above…

A: A) 1+22B) 1-22C) 1+21+2D) 1+21-2E) 22+22

Q: The number 42 has the prime factorization 2- 3-7. Thus, 42 can be written in four ways as a product…

A: Factorise  910 inti prine factors and the find the required pairs.

Q: Prove this: -n is a integer; If 6| n then 18| n^2 +3n

A:

Q: What is the number of nonnegative integer solutions of the equation x1 + x2 + x3 + x4 + x5 = 15,…

A:

Q: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

Q: What are the even integers between -9 and 5?

A: An integer k is said to be even if it can be written as k = 2n where n is an integer.The integers…