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Summing Consecutive Numbers

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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Always the Same

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

Sum Equals Product

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

The sum of the numbers 4 and $1 \frac{1}{3}$ is the same as the product of 4 and $1 \frac{1}{3}$;
that is to say $4 +\frac{4}{3} = 4 \times \frac{4}{3}$.

What other numbers have the sum equal to the product?

What relationship must exist between the numbers?