**Question 171 – “A certain…”**

OG Explanation is fine. Alternatively, you could pick numbers and test the answer choices but I think the direct algebra approach is quite reasonable here.

**Question 172 – “If x, y, z….”**

OG Explanation is fine. Note that: “x is a multiple of z” means the same as “z is a factor of x.”

Alternatively, we could just pick numbers. Let z=2, x=4, y=8.

Choice A: (2+4)/2 = 3 INTEGER

Choice B: (2+8)/4 = 2.5 NOT AN INTEGER (we could stop)

Choice C: (4+8)/2 = 6 INTEGER

Choice D: (4(8))/2 = 8 INTEGER

Choice E: (8(2))/4 = 4 INTEGER

A Word of Caution: If by plugging in numbers you got 4 answer choices as INTEGERS, then that does not mean the 5^{th} answer choice is correct. The question says “not necessarily.”

**Question 173 – “Running…”**

OG Explanation is fine. I don’t know another way to do this problem in a way that is not algebra heavy. This is a pretty challenging and laborious problem. If you are pressed for time just guess or make an educated guess: Letting w=1, 1.25 widgets can be produced in 3 days meaning that it takes 2.4 days to make 1 widget. So the question is how long does it take the slower machine to make 2 widgets? If both machines were working at the same pace, it would take 4.8 days to make 1 widget or 9.6 days to make 2 widgets. Since machine X is slower, it must take machine X longer than 9.6 days. Only answer choices D and E are left. You are down to a 50-50; guess and move on.

**Question 174 – “The product…”**

OG Explanation is okay. Going directly to the choices:

Choice A: The square must be a different digit than the triangle, so this can’t be correct.

Choice B: 12 x 21 will end in a 2 not a 1. Eliminate it.

Choice C: 13 x 31 will end in a 3 not a 1. Eliminate it

Choice D: 21 x 12 will end in a 2. Keep it

Choice E: 31 x 13 will end in a 3. Keep it.

Choice D: 21(12) = 20(12)+12 = 252 Works

Choice E: 31(13) = 30(13)+13 = 403, Does not work.

The correct answer is D.

**Question 175 – “A square…”**

OG Explanation is fine.

**Question 176 – “2 3/5…”**

OG Explanation is fine.