Without Chinese Remainder Theorem

Question

I was helping a young lady study for the (general) GRE. She was using some study guide she had bought. The math sections are divided into "Arithmetic", "Algebra", "Geometry", "Data Analysis." The Arithmetic sections is pre-high school stuff: Simplify exponents, calculate percents, and such.

At the end of the arithmetic section, was a problem: When a positive integer $n$ is divided by $3$ the remainder is $2.$ When it is divided by $5$, the remainder is $1$. Find the smallest possible value of $n$.

I couldn't come up with a good way for her to work the problem. At this level, she wouldn't put division in the shape of the Division Algorithm, so it took some explaining just to get to "Well, $n =3k+2$ for some $k$."

With the two facts $n=3k+2$ and $n=5m+1$, the best I could do was have her write down some terms of both progressions and discover that $11$ was common to both.

So my question is: Is there some technique from pre-algebra (why is that a thing?) that students typically learn which would help them with this problem? Or is it typical of the GRE to ask a question where the student would have to reason as we did (that is, just listing a few terms)? Or is the study guide just whacked?

Answer 1

Another way (not sure if simpler than listing) is to express it as a fraction: $$k=\frac{n-2}{3}; m=\frac{n-1}{5} \Rightarrow k-m=\frac{2n-7}{15}=\frac{2n+8-15}{15}=\frac{2(n+4)}{15}-1.$$ The smallest positive $n=11$.

Answer 2

Calculate backwards:

Dividing by $3$ means subtracting $3$ from $n$ as much as you can until you can't subtract it any more, i.e. when you obtain a number less than $3$. So the next to last step, you obtained $2+3$, the step before, $2+3+3$, and so on.

Therefore the smallest $n$ which satisfies these conditions is the first number in the sequence $\;(2, \,2+3,\,2+3+3,\,…)$ which $1$ more than a multiple of $2$.

Alternatively, you write simultaneously the sequences \begin{align} &2, \,2+3=5,\,2+3+3=8,\,…\\ &1,\,1+5=6,\,1+5+5=11,\,… \end{align} until you find a common term.

Answer 3

The answer must work for 3n+2 and 5m+1, not 3n+1 as inferred in two of the comments.

3*1+2=5, 5*1+1=6;

3*2+2=8, 5*2+1=11;

3*3+2=11

So, 11 is the smallest answer. The multiplier for 3 and 5 does not have to be the same.

Answer 4

The way children are usually taught is:

A. Find the smallest number that has remainder 2 when divided by 3: This is 2. Find more such numbers, and make a list. $$2,5,8,11,...$$

B. Find the smallest number that has remainder 1 when divided by 5: This is 1. Find more such numbers, and make a list: $$1,6,11,...$$

Find the smallest number that goes in both lists.

I have seen this kind of question in fifth grade Math competitions, including upto three numbers, fifth such number, smallest three digit number. It is difficult for young minds, but nevertheless possible with practice.

Do not introduce abstraction. Focus on the numbers themselves.

Answer 5

She could notice that if $x$ is a such number then $x+4$ is divisible by both $3$ and $5$ and it's easy to see that the smallest such number is $15$ so $x=15-4=11$.