Depth · one concept from first principles · step 2 of 4
When each of the four numbers , , , and is divided by the same natural number, all four remainders are equal. Find every possible divisor, together with the remainder for that divisor. (The remainder is not .)
If the divisor is G and the common remainder is r, each number is a multiple of G plus r. The difference of two numbers loses the r.
G divides the difference of any two of the numbers.
7.EE.A.1Convert To AlgebraThe divisor divides the difference of any two of the four numbers.
Each number is some full groups of the divisor plus the same leftover .
Both numbers carry the same extra , and shifting two numbers by the same amount does not change the gap between them, so the difference is just the difference of the full-group parts.
groups of minus groups of is groups of , which is a multiple of .
Subtract neighboring numbers. Every other difference is a sum of these, so these three are enough.
G is a common divisor of 370, 740, 518.
6.NS.B.4Change Focus Count The ComplementFactor each difference. The greatest common factor is 2 × 37 = 74, so the common divisors are the divisors of 74.
G can be 1, 2, 37, or 74.
6.NS.B.4Make A Systematic ListDivide 985 by each candidate. Dividing by 1 always leaves 0, which is not allowed.
Divisor 74 leaves 23, divisor 37 leaves 23, divisor 2 leaves 1.
6.NS.B.4Make A Systematic ListWhen remainders are equal, subtract -- the remainder vanishes and the divisor is left in every difference.
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The same shape shows up later on the AMC, as Small-Domain Constrained Search. Why the shapes repeat →