Based on the information below, what numbers should replace A-D? Everything you need to know is there.

A, B, C, D are all integers. A and B are consecutive integers. B and C, C and D are not consecutive integers.

A < B < C < D

AB + CD = _ _

B x C x D ≠ A x B x C x D = _ 1 _

The first 4 prime numbers – 2, 3, 5, 7!

How'd I deduce that?

I didn't just blindly guess the first 4 prime numbers 🙂 Here's how:

It immediately seemed strange that the multiplication you showed $$(A\times B\times C\times D)$$ uses the $$\times$$ symbol (or rather an x).

There must be reason why – after thinking, it could be because $$AB$$ and $$CD$$ are used to denote $$\overline{AB}$$, or $$A$$ and $$B$$ conjoined to form a 2-digit number. Because of this, $$A,B,C,D$$ must all be single-digit numbers!

From here, I worked out the answer:

As we know $$A,B$$ are consecutive and $$B,C$$ aren't, $$C$$ must be at least $$A+3$$. Why's that important? 🤔

Well, it has to do with how $$\overline{AB}+\overline{CD}$$ is a 2-digit number. If $$A+C$$ is $$9$$ or more, then $$B+D$$ is even larger than 9 – that'd mean $$\overline{AB}+\overline{CD}$$ has 3 digits, and we can safely say $$A+C<9$$.

Now, if $$A=3$$, the minimum possible value of $$C$$ would be $$6$$, and that doesn't meet our criteria. Since $$A\neq1$$, that leaves us with only one possible value for $$A$$: $$2$$. This also means that $$B=3$$!

Also, note that because $$A+C<9$$, $$C$$'s maximum value here is $$6$$.

We can now focus on the last equation:

$$A\times B\times C\times D$$ fits the form $$\_1\_$$. Since $$A=2$$ and $$B=3$$, this is basically $$6\times C\times D$$. All we have to do is test the $$C$$ :)

$$C=6$$ makes it $$36\times D$$, but that yields no results for $$D$$ as a single-digit number. $$36\times8=288$$, and $$36\times9=324$$.

$$C=5$$ must be the case! So, $$30\times D$$ must have $$1$$ as its second digit, and we can quickly find that $$D=7$$.

Also, this is my first answer on this site. I hid the answers because everyone else does, hopefully that's good :)

• Great deduction! Nice first answer, well done :) May 15 at 11:48
• @Prim3numbah thanks, nice puzzle too! Very well-crafted 🙂 May 15 at 13:14

The numbers are:

A=2, B=3, C=5, D=9