Number System

A girl wants to plant trees in her garden in rows in such a way that the number of trees in each row to be the same. There are 10 rows and the number of trees in each row is 12, what is the number of trees in each row, if there are 5 more rows?

Which of the following is the smallest among \( (14)^{\frac{1}{3}},\,(12)^{\frac{1}{2}},\,(16)^{\frac{1}{6}} \) & \( (25)^{\frac{1}{12}} \)?

During a division, Pranjal mistakenly took as the dividend a number that was 10% more than the original dividend. He also mistakenly took as the divisor a number that was 25% more than the original divisor. If the correct quotient of the original division problem was 25 and the remainder was 0, what was the quotient that Pranjal obtained, assuming his calculations had no error?

Which of the following statements is correct?
I. The value of \(100^{2}-99^{2}+98^{2}-97^{2}+96^{2}-95^{2}+94^{2}-93^{2}+\ldots+22^{2}-21^{2}\) is 4840.
II. The value of \(\bigl(K^{2}+\frac{1}{K^{2}}\bigr)\bigl(K-\frac{1}{K}\bigr)\bigl(K^{4}+\frac{1}{K^{4}}\bigr)\bigl(K+\frac{1}{K}\bigr)\bigl(K^{4}-\frac{1}{K^{4}}\bigr)\) is \(K^{16}-\frac{1}{K^{16}}\).

Two numbers, when divided by a certain divisor, leave the remainder 57. When sum of the two numbers is divided by the same divisor, the remainder is 49. The divisor is:

In a division sum, the divisor is 11 times the quotient and 5 times the remainder. If the remainder is 44, then the dividend is:

The six-digit number 7 x 1 yyx is a multiple of 33 for non-zero digits x and y. Which of the following could be a possible value of (x+y)?

The remainder of the term \( 9+9^{2}+\ldots +9^{(2 n+1)} \) when divided by 6 is:

If the seven-digit number 52A6B7C is divisible by 33, and \(A, B, C\) are primes, then the maximum value of \(2A+3B+C\) is:

What is the total number of factors of the number 720 except 1 and the number itself?

What is the least value of \( x+y \), if 10 digit number \( 780 \times 533 \mathrm{y} 24 \) is divisible by 88 ?

A six-digit number 11p9q4 is divisible by 24. Then the greatest possible value for pq is:

A 10-digit number 780533y24 is divisible by 88. What is the smallest possible sum of digits x and y if the last two digits form a number divisible by 4?

Consider a number 780533y24. If this number is divisible by 88, what must be true about the sum of its digits?

A 10-digit number 780533y24 is divisible by 88. Which of the following conditions must be satisfied for y?