Operations on Numbers Questions and Answers

Answer: D

A number is divisible by 80 if it is divisible by both 10 and 8.

For the number to be divisible by 10, its last digit must be 0. So, \(y=0\).

The number now is \(653x0\).

For the number to be divisible by 8, the number formed by its last three digits must be divisible by 8. So, \(3x0\) must be divisible by 8.

Let's check values for x:

If x=2, the number is 65320. \(65320 \div 80 = 816.5\) (Not divisible).

If x=6, the number is 65360. \(65360 \div 80 = 817\) (Divisible).

So the number is 65360, which means \(x=6\) and \(y=0\).

The value of \(x+y = 6+0 = 6\).

Answer: A

We use the formula for the sum of the cubes of the first n natural numbers: \(S_n = [\frac{n(n+1)}{2}]^2\).

The required sum is (Sum of cubes from 1 to 10) - (Sum of cubes from 1 to 4).

Sum of cubes from 1 to 10 = \([\frac{10(10+1)}{2}]^2 = (\frac{110}{2})^2 = 55^2 = 3025\).

Sum of cubes from 1 to 4 = \([\frac{4(4+1)}{2}]^2 = (\frac{20}{2})^2 = 10^2 = 100\).

The required sum = \(3025 - 100 = 2925\).

Answer: B

We only need to find the unit digit of each term and add them.

Unit digit of \(11^1\) is 1.

Unit digit of \(12^2\) is the unit digit of \(2^2\), which is 4.

Unit digit of \(13^3\) is the unit digit of \(3^3=27\), which is 7.

Unit digit of \(14^4\) is the unit digit of \(4^4=256\), which is 6.

Unit digit of \(15^5\) is 5.

Unit digit of \(16^6\) is 6.

Sum of unit digits = \(1+4+7+6+5+6 = 29\).

The unit digit of the final expression is the unit digit of 29, which is 9.

Answer: D

Let's find the unit digits of the first few factorials.

\(1! = 1\) (unit digit is 1)

\(2! = 2\) (unit digit is 2)

\(3! = 6\) (unit digit is 6)

\(4! = 24\) (unit digit is 4)

\(5! = 120\) (unit digit is 0)

\(6! = 720\) (unit digit is 0)

For any n >= 5, n! will have 5 and 2 as factors, so its unit digit will be 0.

So we only need to sum the unit digits of the first four factorials.

Sum of unit digits = \(1 + 2 + 6 + 4 = 13\).

The unit digit of the entire sum is the unit digit of 13, which is 3.

Answer: A

\[ 2x + 9 = 33 \quad\Longrightarrow\quad 2x = 24 \]

\[ x = 12 \]

Answer: A

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Any number with a unit digit of 2 (except for the number 2 itself) is an even number greater than 2.

All even numbers greater than 2 are divisible by 2, and therefore cannot be prime numbers.

Since we are looking for two-digit numbers (like 12, 22, 32,...), all of them will be even and greater than 2. Thus, none of them can be prime.

The number of such primes is 0.

Answer: A

Convert the fractions to decimals to compare them easily.

\(2/3 \approx 0.667\)

\(4/7 \approx 0.571\)

\(5/9 \approx 0.555\)

\(1/2 = 0.500\)

Arranging the decimals in ascending order: 0.500, 0.555, 0.571, 0.667.

The corresponding fractions are: 1/2, 5/9, 4/7, 2/3.

Answer: B

First, find the total fraction of the pizza eaten.

Jane eats 1/4, which is equivalent to 2/8.

Total eaten = Fraction John ate + Fraction Jane ate = \(3/8 + 2/8 = 5/8\).

The whole pizza is 1, or 8/8.

Fraction left = \(8/8 - 5/8 = 3/8\).

Answer: B

Let the number to be subtracted be x.

The new fraction is \(\frac{15-x}{19-x}\).

We are given that this equals 3/4.

\(\frac{15-x}{19-x} = \frac{3}{4}\)

Cross-multiply: \(4(15-x) = 3(19-x)\).

\(60 - 4x = 57 - 3x\).

\(60 - 57 = 4x - 3x\).

\(3 = x\).

Answer: C

Let the required sum be S.

\(S = 2^2 + 4^2 + 6^2 + ... + 20^2\).

\(S = (2 \times 1)^2 + (2 \times 2)^2 + (2 \times 3)^2 + ... + (2 \times 10)^2\).

\(S = 2^2(1^2) + 2^2(2^2) + 2^2(3^2) + ... + 2^2(10^2)\).

\(S = 2^2 (1^2 + 2^2 + 3^2 + ... + 10^2)\).

\(S = 4 \times (\text{given sum})\).

\(S = 4 \times 385 = 1540\).