KA-0122Back-solve from the answer choices
4 points, difficulty 2 of 3Level 5–6about 90sWhich of these numbers has exactly 6 different whole-number factors, counting 1 and the number itself?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
List the factors of each choice in pairs.
2The strategy
The answer is one of the five, so testing beats deriving. For each one, list its factors in pairs that multiply to the number - the pairing stops you missing any.
3The full solution
12 pairs up as 1 and 12, 2 and 6, 3 and 4: six factors. 15 has four, 16 has five, 25 has three, 27 has four.
Solution— no shortcut on this one
Test each choice by listing its factors in pairs. 12: 1 x 12, 2 x 6, 3 x 4 - six factors. 15: 1 x 15, 3 x 5 - four. 16: 1 x 16, 2 x 8, 4 x 4 - five, because 4 pairs with itself. 25: 1 x 25, 5 x 5 - three. 27: 1 x 27, 3 x 9 - four. Only 12 has six. The transferable idea: on a multiple-choice paper, testing five candidates is often faster than finding a rule, and pairing the factors makes each test quick and safe.
Why this is on the test: Back-solving beats deriving here, and pairing factors is what stops a square number being miscounted.