How many two-digit numbers yield a remainder of 1 when divided by both 4 and 14?

A. 0
B. 1
C. 2
D. 3
E. 4

(D) Let’s use n to denote a two-digit number that fits the requirement in the question.

Since we are looking for a remainder of 1 when n is divided by 4 or 14, then (n – 1) must be divisible by both 4 and 14. All numbers divisible by both 4 and 14 must be divisible by their least common multiple, which is 28.

So n – 1 can equal any two-digit multiple of 28. These possible values are:
n – 1 = 28, 56, or 84.

Therefore, n = 29, 57, or 85 (3 different two-digit numbers).

The correct answer is choice (D).
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43 is also another number that yeild a remainder of 1 when divided by both 4 and 14. So the answer should be D.

43 = 40 + 3 = 4 × 10 + 3

When we divide 43 by 4 the remainder is 3, NOT 1.

15 does NOT yield 1 as a remainder, when it is divided by 4.
15 = 3 × 4 + 3

99 = 4 × 24 + 3
So 99 does not give 1 as a remainder when divided by 4.