Question

The remainder when \(9^{19}+7^{49}\) is divided by 64 is (1) 24 (2) 8 (3) 16 (4) 38

Short answer

Answer: 32

Step-by-step solution

Calculate powers for each modulo separately

We will first compute the powers of each term separately using modulo 64. \(9^{19} \mod{64}\) \(7^{49} \mod{64}\)

Calculate powers using the property of exponentiation in modulos

Using the property of exponentiation (a^(m+n) ≡ (a^m) * (a^n)), we can simplify our powers. For \(9^{19} \mod{64}\): \(9^{19} \equiv (9^9)^2 \mod{64}\) \(9^9 \mod{64}\) Similarly, for \(7^{49} \mod{64}\): \(7^{49} \equiv (7^9)^{5}*7^4 \mod{64}\) \(7^9 * 7^4 \mod{64}\)

Calculate the modulo for the simplified powers

Now we calculate the modulo for the powers we have simplified. These calculations should be simple enough that the remainder of the division is easy to determine. \(9^9 \mod{64} = 9\) Therefore, \((9^9)^2 \mod{64} = 9^2 \mod{64} = 81 \mod{64} = 17\) \(7^9 \mod{64} = 15\) \(7^4 \mod{64} = 2401 \mod{64} = 1\) Therefore, \(7^{49} \mod{64} = 15 * 1 \mod{64} = 15\)

Add the modulus of the two powers and calculate the remainder

Now we can add the modulus of the two powers: \(9^{19} + 7^{49} = 17 + 15 = 32\) To obtain the final remainder, we then divide by 64: \(32 \mod{64} = 32\) The remainder is 32, but this option is not listed in the options provided. This might indicate a slight error in the problem or a mistake in the calculation process. Nonetheless, the remainder when \(9^{19} + 7^{49}\) is divided by 64 is 32.

Key concepts

Modular Arithmetic

Modular arithmetic is often referred to as "clock arithmetic" because it deals with numbers wrapping around after reaching a certain value, much like a clock resets after reaching 12. In mathematics, when we say we take a number MOD or modulo another number, our focus is on finding the remainder left over after division.

For instance, if we want to calculate the remainder of 14 when divided by 3, we say 14 mod 3, which equals 2. This is because 3 goes into 14 four times, with a remainder of 2. In problems involving large powers, like the current exercise, modular arithmetic simplifies calculations by focusing only on the remainders.

This is crucial in number theory and many computer algorithms because it lets us work with huge numbers by keeping track of just the remainder, avoiding unnecessary complexity.

Exponentiation

Exponentiation refers to the operation of raising a number to a power. In simpler terms, it's the process of multiplying a number by itself a certain number of times. For example, raising 3 to the power of 4 (expressed as \(3^4\)) involves multiplying 3 by itself for a total of four times, resulting in 81.

When dealing with very large exponents, like \(9^{19}\) or \(7^{49}\), calculations can become daunting. However, using the laws of exponents, such as \(a^{m+n} = a^m \times a^n\), allows us to break down and simplify even the most complex exponent problems. For instance, calculating a large power can be turned into smaller, more manageable calculations, which is especially helpful when brought into the context of modular arithmetic.

Exponentiation is foundational in many areas of mathematics, such as cryptography and computational algorithms.

Remainders

The remainder is what is left after division when one number doesn't divide the other perfectly. For example, dividing 5 by 3 gives a quotient of 1 and a remainder of 2. In many mathematical problems, particularly those involving modular arithmetic, determining the remainder is crucial.

In this exercise, finding the remainder was essential to solving the problem of dividing \(9^{19} + 7^{49}\) by 64. We calculated the remainders for each term separately before adding them:

• The remainder of \(9^{19} \mod{64}\) was 17.
• The remainder of \(7^{49} \mod{64}\) was 15.

Adding these gave us 32, which is the remainder of the entire expression when divided by 64. Understanding remainders helps simplify complex arithmetic problems by making calculations more manageable through focusing on smaller numbers.

Division

Division is one of the fundamental operations in arithmetic, allowing one to split a number into equal parts. It can be thought of as repeated subtraction. When a number is divided by another, it sometimes breaks evenly, and sometimes it leaves a remainder.

In the context of this exercise, division is used within the modular arithmetic framework to find how numbers relate under a mod. Specifically, we were dividing large exponentiated numbers to find their remainders modulo 64. The division was applied when summing the results of \(9^{19} \mod{64}\) and \(7^{49} \mod{64}\) to determine the remainder when everything is combined.

Division, along with multiplication, addition, and subtraction, forms the basis of many more complex mathematical operations and is critical in simplifying expressions in modular arithmetic.

Integer Powers

Integer powers, often just called powers, involve raising a base number to an exponent, both of which are integers. This operation is fundamental in mathematics, allowing us to express repeated multiplication succinctly.

In problems like \(9^{19}\) or \(7^{49}\), integer powers take center stage. To compute these manually can be challenging due to their size, but through properties like \(a^m \cdot a^n = a^{m+n}\), they become more manageable, especially when narrowed down using modular arithmetic.

These properties reduce computational complexity, especially with very large numbers, by allowing the breakdown of calculations into smaller, easier steps. Integer powers are used in various fields from simple algebra to advanced physics, demonstrating their versatility and importance.