Question

Let z be the product of any 5 consecutive three digit odd natural numbers. When you multiply

Answer 1

Answer:

  5

Step-by-step explanation:

If x is the middle number of the 5 consecutive odd numbers, their product is ...

  x(x -2)(x +2)(x -4)(x +4) = x(x² -4)(x² -16)  = x⁵ -20x³ +64x

The first 3-digit prime is 101, so the product of that with this is ...

  101(x⁵ -20x³ +64x) = 101x⁵ -2020x³ +6464x

Since we're only interested in the units digit, we can find the mod 10 value of this to be ...

  (x⁵ +4x) mod 10

Possible units digits of x are 1, 3, 5, 7, 9, so possible units digits of this product are ...

  x(x⁴ +4) = 1(1⁴ +4) = 5

  3(3⁴ +4) = 3(1 +4) = 5 . . . . mod 10

  5(5⁴ +4) = 5(5 +4) = 5 . . . . mod 10

  7(7⁴ +4) = 7(1 +4) = 5 . . . . mod 10

  9(9⁴ +4) = 9(1 +4) = 5 . . . . mod 10

In every case, the units digit of 101z will be 5.