Arithmetic and Number Properties Flashcards

When the positive integer 'n' is divided by 8, the remainder is 5. What is the remainder when 5n is divided by 8?

Answer: 1

If 'n' divided by 8 has a remainder of 5, we can express 'n' as n = 8k + 5 for some integer k. Multiplying by 5 gives 5n = 5(8k + 5) = 40k + 25. The term 40k is perfectly divisible by 8. We then need to find the remainder of 25 when divided by 8. 25 = 3 * 8 + 1. The remainder is 1.

If 'a' is an odd integer and 'b' is an even integer, which of the following expressions must be an even integer?

Answer: a + b + 1

Let's analyze the properties: An odd integer plus an even integer results in an odd integer. An odd integer multiplied by an even integer results in an even integer. A) odd + even + 1 = odd + 1 = even. This must be even. B) (odd * even) + 1 = even + 1 = odd. C) odd - even = odd. D) (3 * odd) + even = odd + even = odd.

What is the units digit of 13^123?

Answer: 7

The units digit of 13^123 is determined by the units digit of the base, which is 3. The pattern of the units digits for powers of 3 is cyclical: 3^1=3, 3^2=9, 3^3=27 (ends in 7), 3^4=81 (ends in 1), 3^5=243 (ends in 3). The cycle is 3, 9, 7, 1, and it repeats every 4 powers. To find the units digit for 13^123, we divide the exponent 123 by the cycle length 4. 123 ÷ 4 = 30 with a remainder of 3. The remainder of 3 corresponds to the third digit in the cycle, which is 7.

The sum of 7 consecutive odd integers is 119. What is the value of the greatest of these integers?

Answer: 23

The average of a set of consecutive integers is the sum of the integers divided by the count. The average is 119 / 7 = 17. For an odd number of consecutive terms, the average is equal to the middle term. Therefore, the middle (4th) integer is 17. The 7 consecutive odd integers are 11, 13, 15, 17, 19, 21, 23. The greatest of these is 23.

The number N is equal to 2^4 * 3^2 * 7^1. How many distinct positive factors does N have?

Answer: 30

To find the total number of distinct positive factors of an integer, first find its prime factorization. For N = p1^a1 * p2^a2 * ... * pk^ak, the total number of factors is (a1 + 1)(a2 + 1)...(ak + 1). Here, the exponents are 4, 2, and 1. So, the number of factors is (4 + 1)(2 + 1)(1 + 1) = 5 * 3 * 2 = 30.

The product of two positive integers, x and y, is 240. If the ratio of x to y is 3 to 5, what is the value of x + y?

Answer: 32

We are given two equations: xy = 240 and x/y = 3/5. From the ratio, we can let x = 3k and y = 5k for some positive integer k. Substituting these into the product equation: (3k)(5k) = 240, which simplifies to 15k^2 = 240. Dividing by 15, we get k^2 = 16, so k = 4. Therefore, x = 3k = 3(4) = 12 and y = 5k = 5(4) = 20. The question asks for the value of x + y, which is 12 + 20 = 32.