Real Numbers Te...

If (a³ + k) is divisible by k, where 'a' is a positive integer and 'k' is a prime number, then which of the following is true?

A real number k/150, where 'k' is a three-digit number with digits 2, 7 and 3 in any order, is expressed as a decimal number. Which of the following is true for the number?

There are three watches, out of which one loses 4 seconds every minute and one gains 4 seconds every minute. If all the watches are set at 8 a.m. and an alarm is activated, each watch beeps for a second when the time turns 1 minute in that watch. At what time will they beep together again?

A and B think about two numbers 'a' and 'b'. They say that the sum of their numbers is 270. If the HCF of the two numbers is 18, then what is the maximum number of such pairs which will satisfy these conditions?

Sham had a certain number of chocolates which he wanted to share among students of a school. At first he tried to distribute equally between two students, then among three, four, five and six students, but every time he was left with one chocolate. When he tried to distribute them among 7 students, he succeeded. What is the smallest number of chocolates he could have?

Which of the following statements is correct?

A. If number X = pⁿ × q^m × r and number Y = p^n-1 × q^m+1 × r, then L.C.M(X, Y) = pⁿ × q^m+1 × r
B. Square root of any non-square number would be an irrational number.
C. H.C.F(1970, 218) = H.C.F(218, 8)
D. The product of primes is equal to a composite number.

Find the correct terms which will replace the letters M, N, O, and P in the given sentences.

A. Zero is not included in the set of …M… numbers.
B. The product of the …N… power of each prime factor involved in the numbers is called …O….
C. Square of an irrational number is …P… a rational number.