Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

14 Divisors

What is the smallest number with exactly 14 divisors?

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?

Stage: 3 Short Challenge Level:

The number is less than $100$, but five times the number is greater than $100$, so it must be between $20$ and $100$.

Reversing the digits makes a prime number, so the first digit must be $1$, $3$, $7$ or $9$, as all two-digit prime numbers are odd and not divisible by $5$. The number must be at least 20, so this rules out $1$.

Since the digits add to a prime number, the possibilities are:
First Digit Second Digit
$3$ $2,4,8$
$7$ $4,6$
$9$ $2,4,8$

The number must be one more than a multiple of $3$, so the digits must sum to give one more than a multiple of $3$, as multiples of $3$ have digit sums that are multiples of $3$. This leaves $34$, $76$ and $94$.

The number must have exactly one prime digit, which rules out $94$.

The number must have exactly four factors. $34$ has $1$, $2$, $17$ and $34$, but $76$ has $1$, $2$, $4$, $19$, $38$ and $76$.

Therefore the number is $34$.

This problem is taken from the UKMT Mathematical Challenges.
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