How many positive two-digit numbers yield a remainder of 1 when divided by 4 and also yield a remainder of 1 when divided by 14?
A. 3
B. 4
C. 5
D. 6
E. 7
Answer: A
Source: Veritas Prep
How many positive two-digit numbers yield a remainder of 1 when divided
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Least value of \(n=1\)BTGModeratorVI wrote: ↑Sat Mar 28, 2020 9:55 amHow many positive two-digit numbers yield a remainder of 1 when divided by 4 and also yield a remainder of 1 when divided by 14?
A. 3
B. 4
C. 5
D. 6
E. 7
Answer: A
Source: Veritas Prep
\(1+ \text{lcm } 2\cdot 14=29=\)next least value of \(n\)
Let \(x=\)number of such two-digit numbers
\(29+(x-1)28 < 100\)
\(x < 3.6\)
\(x=3\)
Therefore, A
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When it comes to remainders, we have a nice rule that says:BTGModeratorVI wrote: ↑Sat Mar 28, 2020 9:55 amHow many positive two-digit numbers yield a remainder of 1 when divided by 4 and also yield a remainder of 1 when divided by 14?
A. 3
B. 4
C. 5
D. 6
E. 7
Answer: A
Source: Veritas Prep
If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.
Two-digit number yields a remainder of 1 when divided by 14.
So, the possible values are: 15, 29, 43, 57, 71, 85 and 99
At this point, we have 7 possible values
Two-digit number yields a remainder of 1 when divided by 4.
Examine each of the 7 values and determine which ones yield a remainder of 1 when divided by 4
They are: 15, 29, 43, 57, 71, 85 and 99
So, there are 3 values that satisfy BOTH conditions.
Answer: A
Cheers,
Brent
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The smallest positive integer that is divisible by both 4 and 14 is their LCM, which is 28. Thus 28 + 1 = 29 will yield a remainder of 1 when divided by both 4 and 14. We can add 28 to each previous dividend to obtain additional dividends:BTGModeratorVI wrote: ↑Sat Mar 28, 2020 9:55 amHow many positive two-digit numbers yield a remainder of 1 when divided by 4 and also yield a remainder of 1 when divided by 14?
A. 3
B. 4
C. 5
D. 6
E. 7
Answer: A
Source: Veritas Prep
29 + 28 = 57
57 + 28 = 85
Since the next number will be 85 + 28 = 113, which is a 3-digit number, we can stop at 85. Thus, we have three 2-digit numbers (29, 57, and 85) that will yield a remainder of 1 when divided by both 4 and 14.
Answer: A
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