# Tag Archives: number sense

## Divisible by 7

How many four digit positive integers divisible by have the property that, when the first and last digits are interchanged, the result is a (not necessarily four-digit) positive integer divisible by ? Source: NCTM Mathematics Teacher, October 2006 Solution Is … Continue reading

## TV Game Show

On a television show, a player receives points for answering an easy question and points for a hard one. What is the largest integer that cannot be a contestant’s total score of the game? Source: NCTM Mathematics Teacher, October 2006 … Continue reading

## Multiple of 3

If is an integer, show why must be a multiple of . Source: NCTM Mathematics Teacher, October 2006 Solution which is a rational number. Clearly, for to be an integer must be a multiple of . Answer: Shown in solution

## Largest of Five Integers

A set contains five integers. When distinct elements of this set are added together, two at a time, the complete list of different possible sums is , and . What is the largest of the five integers in the set? … Continue reading

## 2x^2=3y^2

What is the only pair of integers for which twice the square of the first equals three times the square of the second? Source: NCTM Mathematics Teacher, October 2006 Solution We are looking for integers such that . If , … Continue reading

## Least Positive Integer

What is the least positive integer for which the expression simplifies to an integer? Source: NCTM Mathematics Teacher, October 2006 Solution We look for the smallest sum of positive consecutive integers that is a perfect square a perfect square Answer: Alternative solution … Continue reading

## Same Difference

If fifteen distinct whole numbers are randomly selected from to inclusive, determine the probability that two pairs of these numbers will have the same difference. Source:NCTM Mathematic Teacher, January 2006 Solution What are the possible difference values? There are possible … Continue reading

## Numbers Greater Than 5,000,000

Using all the digits and , how many distinct integers greater than can be formed? Source: NCTM Mathematics Teacher, January 2006 Solution The first digit must be a or a . Case 1: _ _ _ _ _ _ Since … Continue reading

## Arithmetic Sequence

Find the values of the digits , and in the four terms of the following arithmetic sequence: . Source: NCTM Mathematics Teacher, January 2006 Solution The four terms can be expressed as The terms of an arithmetic sequence are increasing … Continue reading

## Perfect Squares and Perfect Cubes

List all integers less than that are both perfect squares and perfect cubes. Source: NCTM Mathematics Teacher, January 2006 Solution is a perfect square but not a perfect cube. On the other hand, is a perfect cube but not a … Continue reading